Local Newton nondegenerate Weil divisors in toric varieties
LLOCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORICVARIETIES
ANDRÁS NÉMETHI AND BALDUR SIGURÐSSON
Abstract.
We introduce and develop the theory of Newton nondegenerate local Weil divisors ( X, in toric affine varieties. We characterize in terms of the toric combinatorics of the Newtondiagram different properties of such singular germs: normality, Gorenstein property, or being anCartier divisor in the ambient space. We discuss certain properties of their (canonical) resolution (cid:101) X → X and the corresponding canonical divisor. We provide combinatorial formulae for the delta–invariant δ ( X, and for the cohomology groups H i ( (cid:101) X, O (cid:101) X ) for i > . In the case dim( X,
0) = 2 ,we provide the (canonical) resolution graph from the Newton diagram and we also prove that ifsuch a Weil divisor is normal and Gorenstein, and the link is a rational homology sphere, then thegeometric genus is given by the minimal path cohomology, a topological invariant of the link.
Contents
1. Introduction 12. Toric preliminaries 33. Analytic Weil divisors in affine toric varieties 44. Newton nondegenerate curve singularities 75. Isolated surface singularities 106. Resolution of Newton nondegenerate surface singularities 117. The geometric genus 138. Canonical divisors and cycle 179. Gorenstein surface singularities 1910. The geometric genus and the diagonal computation sequence 2211. Removing B -facets 2912. Examples 35References 361. Introduction . Hypersurface (or complete intersection) germs with nondegenerate Newton principal part con-stitute a very important family of singularities. They provide a bridge between toric geometry andthe combinatorics of polytopes. The computation of their analytic and topological invariants serveas guiding models for the general cases, and also as testing ground for different general conjecturesand ideas.On the other hand, from the point of view of the general classification theorems in algebraic/analyticgeometry and singularity theory, these hypersurface germs are rather restrictive. In particular, itis highly desired to extend such germs to a more general setting. Besides the algebraic/analyticmotivations there are also several topological ones too: one has to create a flexible family, which isable to follow at analytic level different inductive (cutting and pasting procedures) of the topology.For example, the link of a surface singularity is an oriented plumbed 3–manifold associated witha graph. In inductive proofs and constructions it is very efficient to consider their splice or JSJdecomposition. This would correspond to cutting the Newton diagram by linear planes though their
Date : February 8, 2021.The first author was partially supported by NKFIH Grant “Élvonal (Frontier)” KKP 126683. a r X i v : . [ m a t h . AG ] F e b A. NÉMETHI AND B. SIGURÐSSON Q –Cartier) in the ambienttoric variety. Furthermore, at this level we wish to understand/determine several numerical sheaf–cohomological invarints as well.The second level is the toric combinatorics. In terms of this we wish to characterize the aboveanalytic properties and provide formulae for the numerical invariants.The third level appears explicitly in the case of curve and surface singularities. In the case ofsurfaces we construct the resolution graph (as the plumbing graph of the link, hence as a completetopological invariant). It is always a very interesting and difficult task to decide whether the numer-ical analytic invariants can be recovered from the resolution graph. (This is much harder than theformulae via the toric combinatorics: the Newton polytope preserves considerably more informationfrom the structure of the equations than the resolution graph.) In the last part we prove that thegeometric genus of the resolution can be recovered from the graph. This is a new substantial step ina project which aims to provide topological interpretations for sheaf–cohomological invariants, see[23, 25, 21, 22]1.2 . Next we provide some additional concrete comments and the detailed presentation of the sec-tions.After recalling some notation and results from toric geometry, we generalize the notion of a Newtonnondegenerate hypersurface in C r to an arbitrary Weil divisor in an affine toric variety in section 3.These Newton nondegenerate Weil divisors can be resolved using toric geometry similarly as in theclassical case [27], or in a different generalization [4]. In section 4 we consider Newton nondegeneratecurves. In section 5 we provide conditions for Newton nondegenerate surface singularities to beisolated, and in section 6 we generalize Oka’s algorithm [27] to construct a resolution of a Newtonnondegenerate Weil divisor, along with an explicit description of its resolution graph.In section 7, we give a formula for the δ -invariant and dimensions of cohomologies of the structuresheaf on a resolution of a Newton nondegenerate germ in terms of the Newton polyhedron, seetheorem 7.3, whose statement should have independent interest. In particular, this yields a formulafor the geometric genus. In the classical case, this formula was given by Merle and Teissier in [19,Théorème 2.1.1].In section 8, we give a formula for a canonical divisor on a resolution of a Newton nondegenerateWeil divisor, as well as the canonical cycle in the surface case, in terms of the Newton diagram,see section 8. This formula generalizes results of Oka [27, §9]. In the surface case, we also provein section 9 that the Gorenstein property is identified by the Newton polyhedron, theorem 9.6. Asimilar, but weaker, condition implies that the singularity is Q -Gorenstein, but is not sufficient, asshown by an example in remark 9.8.Using the above results, and a technical result verfied in section 11, we generalize a previousresult [25] for the classical case of Newton nondegenerate hypersurface singularities in C , namelythat the geometric genus is determined by a computation sequence, and is therefore topologicallydetermined:1.3. Theorem.
Let ( X, ⊂ ( Y, be a two-dimensional Newton nondegenerate Weil divisor inthe affine toric ambients space Y . Assume that ( X, is normal and Gorenstein, and that its linkis a rational homology sphere. Then the geometric genus p g ( X, equals the minimal path lattice OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 3 cohomology associated with the link of ( X, . In particular, the geometric genus is determined bythe topology of ( X, . Toric preliminaries
In this section, we will recall some definitions and statements from toric geometry. For an intro-duction, see e.g. [13] and [10].2.1 . Let N be a free Abelian group of rank r ∈ N and set M = N ∨ = Hom( N, Z ) , as well as M R = M ⊗ R and N R = N ⊗ R . If σ ⊂ N R is a cone, the dual cone is defined as σ ∨ = { u ∈ M R | ∀ v ∈ σ : (cid:104) u, v (cid:105) ≥ } . We also set σ ⊥ = { u ∈ M R | ∀ v ∈ σ : (cid:104) u, v (cid:105) = 0 } . We will always assume cones to be finitely generated and rational. To a cone σ ⊂ N R we associatethe semigroup S σ , the algebra A σ and the affine variety U σ by setting S σ = σ ∨ ∩ M, A σ = C [ S σ ] , U σ = Spec( A σ ) . A variety of the form U σ is called an affine toric variety . It has a canonical action of the r -torus T r = ( C ∗ ) r .2.2 . A fan (cid:52) in N is a collection of cones in N R satisfying the following two conditions.(i) Any face of a cone in (cid:52) is in (cid:52) .(ii) The intersection of two cones in (cid:52) is a face of each of them.The support of a fan (cid:52) is defined as |(cid:52)| = ∪ σ ∈(cid:52) σ . If τ, σ ∈ (cid:52) and τ ⊂ σ , then we get a morphism U τ → U σ . These morphisms form a direct system, whose limit is denoted by Y (cid:52) and called theassociated toric variety . The actions of T r on the affine varieties U σ for σ ∈ (cid:52) glue together to forman action on Y (cid:52) . Note that the canonical maps U σ → Y (cid:52) are open inclusions (note also that thenotation Y (cid:52) differs from [13]).Let (cid:101) (cid:52) be another fan in a lattice (cid:101) N and let φ : (cid:101) N → N be a linear map. Assume that for any (cid:101) σ ∈ (cid:101) (cid:52) there is a σ ∈ (cid:52) so that φ ( (cid:101) σ ) ⊂ σ . This induces maps U (cid:101) σ → U σ → Y (cid:52) , which glue togetherto form a map Y (cid:101) (cid:52) → Y (cid:52) .2.3. Lemma (Proposition, p. 39, [13]) . Let (cid:101) (cid:52) and (cid:52) be fans as above. The induced map Y (cid:101) (cid:52) → Y (cid:52) is proper if and only if φ − ( |(cid:52)| ) = | (cid:101) (cid:52)| . . For any p ∈ M , there is an associated rational function on U σ . These glue together to form arational function x p on Y (cid:52) . We refer to these functions as monomials . A monomial x p is a regularfunction on Y (cid:52) if p ∈ |(cid:52)| ∨ = ∩ σ ∈(cid:52) σ ∨ . A map φ : (cid:101) N → N as above induces φ ∗ : M → (cid:102) M . Themonomial x p on Y (cid:52) then pulls back to x φ ∗ ( p ) .2.5 . For σ ∈ (cid:52) , let O σ be the closed subset of U σ defined by the ideal generated by monomials x p where p ∈ ( σ ∨ \ σ ⊥ ) ∩ M . We identify this set with its image in Y (cid:52) . The closure of O σ in Y (cid:52) isdenoted by V ( σ ) . In the case when σ is a ray, V ( σ ) is a Weil divisor and we write D σ = V ( σ ) . Theorbits of the T r action on Y (cid:52) are precisely the sets O σ for σ ∈ (cid:52) . Furthermore, we have (as sets) U σ = (cid:97) τ ⊂ σ O τ , V ( σ ) = (cid:97) σ ⊂ τ O τ , O σ = V ( σ ) \ (cid:91) σ (cid:40) τ V ( τ ) . Let N σ be the subgroup of N generated by σ ∩ N and define N ( σ ) = N/N σ , M ( σ ) = σ ⊥ ∩ M, M σ = M/M ( σ ) . Note that this way we have M σ ∼ = N ∨ σ and M ( σ ) ∼ = N ( σ ) ∨ . Let π σ : N R → N R ( σ ) be the canonicalprojection and set Star( σ ) = { π σ ( τ ) | σ ⊂ τ ∈ (cid:52)} . This set is a fan in N ( σ ) , whose associated toric variety is identified canonically with the orbit closure V ( σ ) . Similarly, let (cid:36) σ : M → M σ be the canonical projection. Assuming σ ∈ (cid:52) has dimension s , A. NÉMETHI AND B. SIGURÐSSON we have ( U σ , O σ ) ∼ = ( Y σ × ( C ∗ ) r − s , ( { } × ( C ∗ ) r − s ) . In particular, O σ ⊂ Y (cid:52) has Y σ as a transversetype.2.6. Definition. (i) For a cone Σ ⊂ N R , let (cid:52) Σ denote the fanfan consisting of all the faces of Σ .We also write Y Σ instead of Y (cid:52) Σ .(ii) If (cid:52) is a fan and i ∈ N , define (cid:52) ( i ) = { σ ∈ (cid:52) | dim σ = i } . (iii) A regular cone (resp. simplicial cone ) is a cone generated by a subset of an integral (resp.rational) basis of N .(iv) A subdivision of a fan (cid:52) is a fan (cid:101) (cid:52) so that | (cid:101) (cid:52)| = |(cid:52)| and each cone in (cid:52) is a union of conesin (cid:101) (cid:52) . A regular subdivision is a subdivision consisting of regular cones.(v) If Σ ⊂ N R is a cone and (cid:52) is a subdivision of (cid:52) Σ , denote by (cid:52) ∗ the fan consisting of σ ∈ (cid:52) forwhich σ ⊂ ∂ Σ . Here we see ∂ Σ as the union of the proper faces of Σ . As a result, (cid:52) ∗ is a subdivisionof the fan (cid:52) Σ \ { Σ } .(vi) Let (cid:52) , (cid:52) be subdivisions of a fan (cid:52) . We say that (cid:52) refines (cid:52) if (cid:52) is a subdivision of (cid:52) ,or that (cid:52) is a refinement of (cid:52) .(vii) Let (cid:52) be a fan with a subdivision (cid:52) and let σ ∈ (cid:52) . The restriction of (cid:52) to σ is defined as (cid:52) | σ = { τ ∈ (cid:52) | τ ⊂ σ } . Analytic Weil divisors in affine toric varieties . Throughout this section, as well as the following sections, we will assume that N has rank r andthat Σ is an r -dimensional, rational, finitely generated, strictly convex cone in N R . This means that Σ ⊂ N R is generated over R ≥ by a finite set of elements from N , which generate N as a vectorspace,and that Σ ⊥ = { } . In particular, the orbit O Σ consists of a single point, which we denote by ,and refer to as the origin . Let Y Σ be the affine toric variety associated with Σ .Any subdivision (cid:52) of (cid:52) Σ induces a modification π (cid:52) : Y (cid:52) → Y Σ .In the sequel we denote by ( Y Σ , the analytic germ of Y Σ at 0, and usually we will denote by Y a(small Stein) representative of ( Y Σ , . (Hence ( Y,
0) = ( Y Σ , .) If π ∆ is a toric modification, in thediscussions regarding the local analytic germ ( Y, , we will use the same notation Y ∆ for π − ( Y ) and D σ for D σ ∩ π − ( Y ) . Similarly, O σ might stay for O σ ∩ Y ⊂ Y as well. If in some argument wereally wish to stress the differences, we write Y loc ∆ , D locσ , O locσ for the local objects.Assume that f ∈ O Y, is the germ of a holomorphic function at the origin, which has an expansion(3.1) f ( x ) = (cid:88) p ∈ S Σ a p x p , a p ∈ C . Then ( { f = 0 } , ⊂ ( Y, is the germ of an analytic space. We set supp( f ) = { p ∈ S Σ | a p (cid:54) = 0 } too.3.2. Definition.
The
Newton polyhedron of f with respect to Σ is the polyhedron Γ + ( f ) = conv(supp( f ) + Σ ∨ ) , where conv denotes the convex closure in M R . The union of compact faces of Γ + ( f ) is denoted by Γ( f ) and is called the Newton diagram of f with respect to Σ .3.3 . The fan (cid:52) f and some combinatorial properties. It follows from definition that Σ isprecisely the set of those linear functions on M R having a minimal value on Γ + ( f ) . Denote by F ( (cid:96) ) the minimal set of (cid:96) ∈ Σ on Γ + ( f ) . For (cid:96) , (cid:96) ∈ Σ , say that (cid:96) ∼ (cid:96) if and only if F ( (cid:96) ) = F ( (cid:96) ) .Then ∼ is an equivalence relation on Σ having finitely many equivalence classes, each of whoseclosure is a finitely generated rational strictly convex cone. These cones form a fan, which we willdenote by (cid:52) f . We refer to (cid:52) f as the dual fan associated with f and Σ . Note that (cid:52) f refines (cid:52) Σ .For any σ ∈ (cid:52) f , the face F ( (cid:96) ) is independent of the choice of (cid:96) ∈ σ ◦ , where σ ◦ ⊂ σ is the relativeinterior, that is, the topological interior of σ as a subset of N σ, R . For σ ∈ (cid:52) (1) f , the set σ ∩ N is asemigroup generated by a unique element, which we denote by (cid:96) σ . For a series g ∈ O Y, [ x M ] = { x p h | p ∈ M, h ∈ O Y, } , OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 5 the support supp( g ) is defined similarly as above, and for σ ∈ (cid:52) (1) f we set wt σ ( g ) = min { (cid:96) σ ( p ) | p ∈ supp( g ) } . One verifies that for any such g (3.2) the vanishing order of g along D σ ⊂ Y (cid:52) f is exactly wt σ ( g ) . Definition.
Let σ ∈ (cid:52) f and (cid:96) ∈ σ ◦ . Define F σ = F ( (cid:96) ) , f σ = (cid:88) p ∈ F σ a p x p . If σ (cid:48) ⊂ N R is a cone, and σ (cid:48)◦ ⊂ σ ◦ (for example, if σ (cid:48) is an element of a refinement of (cid:52) f ), thenwe set F σ (cid:48) = F σ .If σ ⊂ Σ is one dimensional, set m σ = wt σ ( f ) . Thus, (cid:96) σ | F σ ≡ m σ . Note that we have Γ + ( f ) = (cid:110) u ∈ M R (cid:12)(cid:12)(cid:12) ∀ σ ∈ (cid:52) (1) f : (cid:96) σ ( u ) ≥ m σ (cid:111) . This can be compared with the following set.3.5.
Definition.
Let Γ ∗ + ( f ) = (cid:110) u ∈ M R (cid:12)(cid:12)(cid:12) ∀ σ ∈ (cid:52) ∗ (1) f : (cid:96) σ ( u ) ≥ m σ (cid:111) , where (cid:52) ∗ (1) f denotes the set of rays in (cid:52) f contained in the boundary of Σ .3.6. Definition.
Denote by ( X, ⊂ ( Y, the union of those local primary components of thegerm defined by f (with their non-reduced structure), which are not invariant by the torus action.If f is reduced along the non-invariant components, this means the following. Let U ⊂ Y be aneighbourhood of the origin on which f converges and let X (cid:48) ⊂ U be defined by f = 0 . Then X isthe closure of X (cid:48) \ ∪ (cid:110) D σ (cid:12)(cid:12)(cid:12) σ ∈ (cid:52) (1)Σ (cid:111) in U .3.7. Remark. (i) For any p ∈ M , the function x p f defines the same germ ( X, . Thus, we mayallow f ∈ O Y, [ x M ] = { x p g | p ∈ M, g ∈ O Y, } as well.(ii) Since the divisors { D σ : σ ∈ (cid:52) (1)Σ } are torus-invariant, the divisor of f in Y Σ is X + (cid:80) σ m σ D σ .3.8. Proposition. (i) We have Γ + ( f ) = p + Σ ∨ for some p ∈ M if and only if (cid:52) f = (cid:52) Σ if andonly if the germ X at is the empty germ.(ii) For a σ ∈ (cid:52) Σ , we have O σ ⊂ X if and only if the normal fan (cid:52) f subdivides σ into smallercones, i.e. (cid:52) f | σ (cid:54) = (cid:52) σ .(iii) The ideal I X ⊂ O Y, which defines ( X, in ( Y, , is generated by the functions x p f for p ∈ M satisfying (cid:96) σ ( p ) + m σ ≥ for all σ ∈ (cid:52) (1)Σ .Proof. Statement (i) is clear, since Γ + ( f ) is of the form p + Σ ∨ if and only if f is a product of amonomial and a unit in O Y, .Statement (ii) follows from (i), and the fact that the intersection of X and a generic transversespace Y σ to O σ has Newton polygon (cid:36) σ (Γ( f )) , cf. 2.5.(iii) Assuming the given conditions on p , the function x p f is meromorphic and has no poles. Since Y is normal, x p f is analytic and vanishes on X . As a result, x p f ∈ I X .To show that these generate I X , take g ∈ I X . We must show that g = hf , with h ∈ O Y, [ x M ] and (cid:96) σ ( p ) + m σ ≥ for p ∈ supp( h ) .Let I X,M be the localization of I X along the invariant divisors, that is, the ideal of meromorphicfunction germs on ( Y, , regular on the open torus and vanishing on X . It follows that I X,M = f · O Y, [ x M ] and I X = I X,M ∩ O Y, .Thus, g = x r hf for some h ∈ O Y, and r ∈ M . Then, there exist finite families ( h i ) i of units in O Y, and exponents ( p i ) i in M so that x r h = (cid:80) i x p i h i and the support of x r h is the disjoint unionof the supports of x p i h i . Let us take any σ ∈ (cid:52) (1)Σ . The condition on disjointness of supports gives min i wt σ x p i h i f = wt σ x r hf = wt σ g ≥ . As a result, we have (cid:96) σ ( p i ) + m σ ≥ for all i . The result follows. (cid:3) A. NÉMETHI AND B. SIGURÐSSON
Definition.
Let f and (cid:52) f be as above. We say that Γ + ( f ) , or f , is ( Q -) pointed if there existsa p ∈ M ( p ∈ M Q ) such that (cid:96) σ ( p ) = m σ for all σ ∈ (cid:52) (1)Σ .3.10. Proposition. (i) If Σ is regular (resp. simplicial), then any Newton polyhedron (w.r.t. Σ ) ispointed at some p ∈ M (resp. p ∈ M Q ).(ii) f is pointed at p ∈ M if and only if ( X, in ( Y, is defined by a single equation x − p f (cf.proposition 3.8). In other words, f is pointed if and only if ( X, is a Cartier divisor in ( Y, .(iii) f is pointed at p ∈ M Q if and only if ( X, is a Q -Cartier divisor in ( Y, .Proof. (i) Use the fact that { (cid:96) σ : σ ∈ (cid:52) (1)Σ } is an integral (resp. rational) basis.(ii) If f is pointed at p ∈ M then by proposition 3.8, x − p f ∈ I X . Moreover, if x − q f ∈ I X forsome q ∈ M , then (cid:96) σ ( p − q ) ≥ for any σ ∈ (cid:52) (1)Σ , hence p − q ∈ Σ ∩ M and x p − q ∈ O Y, .Conversely, assume that ( X, ⊂ ( Y, is an (analytic) Cartier divisor. Let (cid:101) (cid:52) f be a smoothsubdivision of (cid:52) f , and set (cid:101) Y = Y (cid:101) (cid:52) f . This is a smooth variety, and the map π : (cid:101) Y → Y Σ is aresolution of Y . Take a small Stein representative Y loc ⊂ Y , and set (cid:101) Y loc = π − ( Y loc ) . Then wehave the vanishing H ≥ ( (cid:101) Y , O (cid:101) Y ) = 0 (see e.g. [13, Corrollary, p. 74] or [10, §8.5]), and also its localanalogue H ≥ ( (cid:101) Y loc , O (cid:101) Y loc ) = 0 (since the local analytic germ ( Y, is rational too). Thus, from theexponential exact sequence, Pic( (cid:101) Y ) = H ( (cid:101) Y , Z ) and Pic( (cid:101) Y loc ) = H ( (cid:101) Y loc , Z ) . On the other hand, Y is weighted homogeneous (as any affine toric variety), hence H ( (cid:101) Y , Z ) = H ( (cid:101) Y loc , Z ) . In particular, Pic( (cid:101) Y ) ∼ = Pic( (cid:101) Y loc ) . Here the first group is the Picard group of the algebraic variety, while thesecond one is the Picard group of the analytic manifold.Next, consider the Chow group A r − ( Y ) of codimension one, i.e. the group freely generated byWeil divisors, modulo linear equivalence. Note that since (cid:101) Y is smooth, we have A r − ( (cid:101) Y ) ∼ = Pic( (cid:101) Y ) and A r − ( (cid:101) Y loc ) ∼ = Pic( (cid:101) Y loc ) . If we factor these isomorphic groups by the subgroups generated bythe exceptional divisors, we find that the restriction induces an isomorphism A r − ( Y ) ∼ = A r − ( Y loc ) .Denote by D loc σ the restriction image of D σ under the above isomorphism. Since ( X, ⊂ ( Y, is local analytic Cartier, and the local divisor of f in Y is X + D loc f , where D loc f = (cid:80) σ ∈(cid:52) (1)Σ m σ D loc σ ,we find that the class of D loc f is zero in A r − ( Y loc ) . But then, by the above isomorphisms, the classof D f = (cid:80) σ ∈(cid:52) (1)Σ m σ D σ is zero in A r − ( Y ) .Finally note that A r − ( Y ) can be computed as follows [13, 3.4]. Consider the group Div T ( Y ) = Z (cid:104) D σ | σ ∈ (cid:52) Σ (cid:105) of invariant divisors and the inclusion M (cid:44) → Div T ( Y ) sending p ∈ M to (cid:80) σ (cid:96) σ ( p ) D σ .Along with the map Div T → A r − ( Y ) , this gives a short exact sequence → M → Div T ( Y ) → A r − ( Y ) → . Since D f ∈ A r − ( Y ) maps to zero in A r − ( Y loc ) under the above isomorphism, and D f ∈ Div T ( Y ) ,we find that D f is in the image of M . But this means exactly that there exists p ∈ M such that (cid:96) σ ( p ) = m σ for all σ ∈ (cid:52) (1)Σ .(iii) Use part (ii) for a certain power of f . (cid:3) Definition.
We say that f has Newton nondegenerate principal part with respect to Σ (orsimply that f or ( X, is Newton nondegenerate ) if for every σ ∈ (cid:52) f with F σ compact, the variety Spec( C [ M ] / ( f σ )) (that is, { x ∈ T r | f σ ( x ) = 0 } with its non-reduced structure) is smooth. Note that f σ is a polynomial since F σ is compact.3.12. Lemma.
Assume that ( X, ⊂ ( Y, is Newton nondegenerate and let σ ∈ (cid:52) Σ . If O σ ⊂ X ,then the generic transverse type of X to O σ is a Newton nondegenerate singularity with Newtonpolyhedron (cid:36) σ (Γ + ( f )) ⊂ M σ .Proof. The statement follows by restricting f to a toric subspace transverse to O σ , see 2.5. (cid:3) . The fan (cid:101) (cid:52) f and the associated resolution. Assume that f is Newton nondegenerate. Let (cid:101) (cid:52) f be a regular subdivision of (cid:52) f . Then (cid:101) Y = Y (cid:101) (cid:52) f is a smooth variety, and we have a modification π : (cid:101) Y → Y . As a result of the nondegeneracy of f , the strict transform (cid:101) X of X in (cid:101) Y intersects allorbits in (cid:101) Y smoothly. In particular, (cid:101) X is smooth, and π is an embedded resolution of ( X, ⊂ ( Y, . OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 7
Lemma.
Assume ( X, ⊂ ( Y, is a Newton nondegenerate Weil divisor. Then, the singularlocus of the germ ( X, is contained in the union of codimension ≥ orbits in ( Y, .Proof. Let Y ( ≤ r − be the union of orbits of dimension ≤ r − , that is, codimension ≥ , in Y . Let π : (cid:101) Y → Y be as in 3.13. The restriction π − ( Y \ Y ( ≤ r − ) → Y \ Y ( ≤ r − is an isomorphism, and (cid:101) X is smooth. Therefore, X \ Y ( ≤ r − is smooth. (cid:3) Newton nondegenerate curve singularities
In this section, we will assume that rk N = 2 and that Σ ⊂ N R is a two dimensional finitelygenerated strictly convex rational cone. Nondegenerate rank 2 singularities appear naturally in the r = 3 case as transversal types of certain orbits.We will introduce the canonical subdivision and we establish criterions for irreducibility andsmoothness. They will be used in the context of rank r = 3 cones in the definition of their canonicalsubdivision and in the characterization of Newton nondegenerate isolated surface singularities.4.1 . Canonical primitive sequence. Assume first that Σ is nonregular. Then there exists asequence of vectors (cid:96) , . . . , (cid:96) s +1 ∈ Σ ∩ N , called the canonical primitive sequence [27] and integers b , . . . , b s ≥ , called the associated selfintersection numbers , so that:(i) If ≤ j ≤ s , then (cid:96) j , (cid:96) j +1 form an integral basis for N .(ii) If < j ≤ s , then b j (cid:96) j = (cid:96) j − + (cid:96) j +1 .(iii) The set { (cid:96) , . . . , (cid:96) s +1 } is a minimal set of generators for the semigroup Σ ∩ N .This data is uniquely determined up to reversing the order of ( (cid:96) j ) j and ( b j ) j . It can, in fact, bedetermined as follows. Let α be the absolute value of the determinant of the × matrix whosecolumns (cid:96), (cid:96) (cid:48) are the primitive generators of the one dimensional faces of Σ , given in any integralbasis. Then, there exists a unique integer ≤ β < α so that β(cid:96) + (cid:96) (cid:48) ∈ αN . The selfintersectionnumbers are determined as the negative continued fraction expansion αβ = b − b − . . . − b s . We use the notation [ b , . . . , b s ] for the right hand side above. We have (cid:96) = (cid:96), (cid:96) = β(cid:96) + (cid:96) (cid:48) α . Along with condition (ii), this determines the canonical primitive sequence recursively and we have (cid:96) s +1 = (cid:96) (cid:48) . (cid:96) = (5 , (cid:96) = (3 , (cid:96) = (1 , (cid:96) = (0 , Figure 1.
In this example, Σ is generated by (0 , and (5 , . The canonicalprimitive sequence consists of four elements, including the generators of the cone.Alternatively, the vectors (cid:96) , (cid:96) , . . . , (cid:96) s +1 are the integral points lying on compact faces of theconvex closure of the set Σ ∩ N \ { } . For a detailed discussion of this construction, see [26, 1.6].If Σ is regular, then we prefer to modify the minimality of the resolution considered above, andset s = 1 , (cid:96) = (cid:96) and (cid:96) = (cid:96) (cid:48) and (cid:96) = (cid:96) + (cid:96) . Accordingly, in (ii), we will have − b = − . Inparticular, the set { (cid:96) , (cid:96) , (cid:96) } is not a minimal set of generators of the semigroup Σ ∩ N . We make A. NÉMETHI AND B. SIGURÐSSON this choice here mostly for technical reasons (directed by properties of the induced reslution), whichwill appear in section 10. The same choice is made in [27], Definition (3.5).4.2.
Definition.
Let Σ be a two dimensional rational strictly convex cone with a canonical primitivesequence (cid:96) , (cid:96) , . . . , (cid:96) s +1 . The canonical subdivision of (cid:52) Σ is the unique subdivision (cid:101) (cid:52) Σ for which (cid:101) (cid:52) (1)Σ = { R ≥ (cid:104) (cid:96) i (cid:105) | ≤ i ≤ s + 1 } . For each i = 1 , . . . , s , there is a unique number − b i ∈ Z ≤− satisfying (cid:96) i − − b i (cid:96) i + (cid:96) i +1 = 0 . Wedefine α ( (cid:96) , (cid:96) s +1 ) and β ( (cid:96) , (cid:96) s +1 ) as the numerator and denominator, respectively, of the negativecontinued fraction [ b , . . . , b s ] = b − b − ··· , (we require gcd( α ( (cid:96) , (cid:96) s +1 ) , β ( (cid:96) , (cid:96) s +1 )) = 1 , and β ( (cid:96) , (cid:96) s +1 ) ≥ , so that these numbers are welldefined). The number α ( (cid:96) , (cid:96) s +1 ) is referred to as the determinant of Σ .4.3. Remark.
Let (cid:96) , (cid:96) ∈ N be two linearly independent elements. Then we have α ( (cid:96) , (cid:96) ) = 1 ifand only if (cid:96) , (cid:96) form part of an integral basis of N . In general, α = α ( (cid:96) , (cid:96) ) can be computed asthe content of the restriction of (cid:96) to the kernel of (cid:96) . In other words, let K ⊂ N be the kernel of (cid:96) . Then (cid:96) | K is divisible by α , and ( (cid:96) | K ) /α is primitive.4.4. Lemma. If Σ is not a regular cone, then Y Σ has a cyclic quotient singularity at the origin andthe map Y (cid:101) (cid:52) Σ → Y Σ induced by the identity map on N is the minimal resolution.Proof. See Proposition 1.19 and Proposition 1.24 of [26]. (cid:3)
Proposition.
Assume that rk N = 2 , and that f is Newton nondegenerate with respect to Σ ⊂ N R defining a germ ( X, .(i) The germ ( X, is irreducible if and only if Γ( f ) is a single interval with no integral interiorpoints. In fact, in general, the number of components in ( X, is precisely the combinatoriallength of Γ( f ) .(ii) Assume that ( X, is irreducible and let σ ∈ (cid:52) (1) f so that Γ( f ) = F σ . Then ( X, is smoothif and only if (cid:96) σ lies on the boundary of the convex hull of the set Σ ◦ ∩ N . In other words,let (cid:96) , . . . , (cid:96) s +1 be the canonical primitive sequence of Σ . Then either (cid:96) σ is one of (cid:96) , . . . , (cid:96) s ,or there is an a ∈ Z > such that either (cid:96) σ = a(cid:96) + (cid:96) or (cid:96) σ = a(cid:96) s +1 + (cid:96) s . (iii) The curve ( X, is smooth if and only if the following condition holds: If p ∈ M and (cid:96) σ ( p ) > m σ for all σ ∈ (cid:52) (1)Σ , then (cid:96) σ ( p ) > m σ for all σ ∈ (cid:52) (1) f . Remark.
One can ask why the vectors (cid:96) and (cid:96) s +1 do not appear in the list of (ii). The answeris that the corresponding divisors D σ , though they intersect E transversaly, they are T –invariant,hence they are eliminated by the convention of the definition 3.6. Proof of proposition 4.5.
We start with the following observations. Write σ i = R ≥ (cid:104) (cid:96) i (cid:105) . Let (cid:52) (cid:48) bea regular subdivision of (cid:52) Σ which refines both (cid:52) f and the canonical subdivision of (cid:52) Σ . The map Y (cid:52) (cid:48) → Y Σ is then a resolution of Y Σ with exceptional divisor E (cid:48) . We can write E (cid:48) = ∪ s (cid:48) i =1 E (cid:48) i , whereeach E (cid:48) i is a rational curve. Furthermore, if i (cid:54) = j , then E (cid:48) i and E (cid:48) j intersect if and only if | i − j | = 1 .In fact, we can write (cid:52) (cid:48) (1) = { σ (cid:48) , . . . , σ (cid:48) s (cid:48) } ∪ { σ, τ } where σ, τ are the two faces of Σ and E (cid:48) i = V ( σ (cid:48) i ) .Similarly as in [27], we see that Y (cid:52) (cid:48) resolves ( X, and that the strict transform X (cid:48) of X in Y (cid:52) (cid:48) intersects the exceptional divisor E (cid:48) transversally in smooth points of E (cid:48) . In fact, these intersectionpoints lie in the open orbit O σ (cid:48) i ⊂ E (cid:48) i . Therefore, we have (see [27, Theorem 5.1]) | X (cid:48) ∩ E (cid:48) i | = χ ( X (cid:48) ∩ O σ (cid:48) i ) = Vol ( F ( (cid:96) (cid:48) i )) where (cid:96) (cid:48) i is the primitive generator of σ (cid:48) i . Now, the components of ( X, are in bijection with theintersection points X (cid:48) ∩ E (cid:48) , which proves (i). OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 9
For (ii), let (cid:101) (cid:52) Σ be the canonical subdivision, and π : (cid:101) Y → Y the associated modification, whichis a resolution of Y . Let (cid:101) X ⊂ (cid:101) Y be the strict transform of X . The minimal cycle of the resolution (cid:101) Y → Y is the reduced exceptional divisor E ⊂ (cid:101) Y and ( Y, is rational. By [5], the pullback of themaximal ideal of ∈ Y is the reduced exceptional divisor in (cid:101) Y , and the maximal ideal has no basepoints in (cid:101) Y . It follows that the multiplicity of ( X, is the intersection number between (cid:101) X and E .In particular, ( X, is smooth if and only if E ∪ (cid:101) X is a normal crossing divisor. If σ = σ i for some ≤ i ≤ s , then this is indeed the case. Otherwise, there is an ≤ i ≤ s so that (cid:96) σ = a(cid:96) i + b(cid:96) i +1 .In a neighbourhood of E i ∩ E i +1 we have coordinates u, v so that E i = { x = 0 } , E i +1 = { y = 0 } and we have some generic coefficients c, d so that the strict transform of X is defined by cx b + dy a .Thus, ( X, is not smooth if < i < s . In the case i = 0 (the case i = s is similar), (cid:101) X is smoothand transverse to E if and only if b = 1 .The condition in (iii) is equivalent with the equality(4.1) (Γ ∗ + ( f ) \ Γ + ( f )) ∩ M = ∅ . Choose a basis for N , inducing an isomorphism N ∼ = M via the dual basis, as well as an inner producton N R ∼ = M R . If we rotate the segment Γ( f ) by π/ and translate it, then it can be identified withthe vector (cid:96) i (segment t(cid:96) i , t ∈ [0 , ). Consider the parallelogram P ( (cid:96) i ) whose sides are parallel to (cid:96) and (cid:96) s +1 , and it has (cid:96) i as diagonal. It is divided by (cid:96) i into two triangles, each of them can beidentified by Γ ∗ + ( f ) \ Γ + ( f ) . Hence, eq. (4.1) holds if and only if P ( (cid:96) i ) ◦ ∩ N = ∅ .Clearly, P ( (cid:96) i ) ◦ ∩ N is empty if (cid:96) σ ∈ ∂ conv(Σ ◦ ∩ N ) . The converse can be seen as follows. Let ( (cid:96) b i ) i ∈ Z be a family consisting of integral points on ∂ conv(Σ ◦ ∩ N ) , ordered according one of theorientation of this boundary. Two consecutive elements of this family form a basis of N , and Σ ◦ ∩ N = (cid:91) i ∈ Z Z ≥ (cid:104) (cid:96) b i , (cid:96) b i +1 (cid:105) \ { } . It follows that the set of irreducible elements in the semigroup Σ ◦ ∩ N are presicely the elements onthe boundary ∂ conv(Σ ◦ ∩ N ) . In particular, if (cid:96) σ ∈ (conv(Σ ◦ ∩ N )) ◦ , then (cid:96) σ = (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) for some (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ∈ Σ ◦ ∩ N . It follows that (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ∈ P ( (cid:96) i ) ◦ . (cid:3) conv(Σ ◦ ∩ N ) ⊂ ΣΓ + ( f ) ⊂ Σ ∨ Figure 2.
The integral points in the interior of the parallellogram P ( (cid:96) σ ) .4.7. Corollary.
Consider the notation from the proof of proposition 4.5(ii), that is, ( X, irreducibleand (cid:96) σ = a(cid:96) i + b(cid:96) i +1 with gcd( a, b ) = 1 . Then the multiplicity of ( X, is mult( X,
0) = b i = 0 ,a + b < i < s,a i = s. (cid:3) Remark.
Let (cid:96), (cid:96) (cid:48) be any two linearly independent integral vectors in any free Z module, andlet N be the free Z module generated by them. Then the definitions from 4.1 and 4.2 can be repeatedin N . Then the determinant of two such vectors can be seen as the greatest common divisor of themaximal minors of the matrix having the coordinate vectors of (cid:96), (cid:96) (cid:48) as rows, see [27]. Note that α ( (cid:96), (cid:96) (cid:48) ) = α ( (cid:96) (cid:48) , (cid:96) ) . Moreover, β ( (cid:96) , (cid:96) s +1 ) β ( (cid:96) s +1 , (cid:96) ) ≡ α ( (cid:96) , (cid:96) s +1 )) , cf. [29, Proposotion 5.6]. Isolated surface singularities
In the next theorem we give necessary and sufficient conditions for a Newton nondegeneratesurface singularity to be isolated, in terms of the Newton polyhedron. In particular, we assume that r = 3 in this section. This is a (non-direct) generalization of a result of Kouchnirenko valid in theclassical case [15].5.1. Theorem.
Let ( X, be a Newton nondegenerate singularity and assume rk N = 3 . Thefollowing are equivalent(i) ( X, has an isolated singularity.(ii) If p ∈ M satisfies (cid:96) σ ( p ) > m σ for all σ ∈ (cid:52) (1)Σ , then (cid:96) σ ( p ) > m σ for all σ ∈ (cid:52) ∗ (1) f .(iii) Let σ , σ ∈ (cid:52) (1)Σ and σ = R ≥ (cid:104) σ , σ (cid:105) ∈ (cid:52) (2)Σ and assume that τ ∈ (cid:52) (1) f with τ ⊂ σ . If p ∈ M so that (cid:96) σ ( p ) > m σ and (cid:96) σ ( p ) > m σ , then (cid:96) τ ( p ) > m τ .(iv) Let σ , σ ∈ (cid:52) (1)Σ and σ = R ≥ (cid:104) σ , σ (cid:105) ∈ (cid:52) (2)Σ . Then there is at most one τ ∈ (cid:52) (1) f with τ ⊂ σ and σ (cid:54) = τ (cid:54) = σ . If such a τ exists, then (cid:96) τ is one of the following (5.1) (cid:96) , . . . , (cid:96) s , a(cid:96) + (cid:96) , (cid:96) s + a(cid:96) s +1 , a ∈ Z ≥ and, furthermore, there exists an e ∈ Q so that (5.2) e(cid:96) τ + (cid:96) σ α ( (cid:96) τ , (cid:96) σ ) + (cid:96) σ α ( (cid:96) τ , (cid:96) σ ) = 0 , (see definition 4.2 for α ( · , · ) ) and (5.3) em τ + m σ α ( (cid:96) τ , (cid:96) σ ) + m σ α ( (cid:96) τ , (cid:96) σ ) = − . Proof.
By lemma 3.14, the singular locus of the punctured germ X \ { } is a union of orbits O σ forsome σ ∈ (cid:52) (2)Σ . For such a σ , we have ( V ( σ ) , ⊂ ( X, if and only if the projection of Γ + ( f ) in M σ is nontrivial, by lemma 3.12. By the same lemma, if ( V ( σ ) , ⊂ ( X, , then the generic transversetype to V ( σ ) in ( X, is a Newton nondegenerate curve with Newton polyhedron the projection of Γ + ( f ) to M σ . Therefore (i) ⇔ (iii) follows from proposition 4.5. The equivalence of (ii) and (iii) isan exercise.The generic transverse type to ( V ( σ ) , in ( X, is smooth if and only if its diagram has a singleface corresponding to a τ as in eq. (5.1), and this face has length one. (i) ⇔ (iv) follows, once weshow that given such a τ , an e ∈ Q satisfying eq. (5.2) exists and is unique, and that, furthermore,the left hand side of eq. (5.3) is minus the combinatorial length of the face F of the Newton diagramcorresponding to τ .Take a smooth subdivision of σ containing τ as a ray, and let τ i be the ray adjacent to τ between τ and σ i . Then there exists a − b ∈ Z so that(5.4) − b(cid:96) τ + (cid:96) τ + (cid:96) τ = 0 . Furthermore, for i = 1 , , we may assume that (cid:96) τ i = β i (cid:96) τ + (cid:96) σ i α i where α i /β i is the continued fraction associated with (cid:96) τ and (cid:96) σ i . As a result, eq. (5.4) can berewritten as (5.2) with e = − b + β /α + β /α . Let p , p be the endpoints of F so that (cid:96) τ ( p − p ) > and (cid:96) τ ( p − p ) > . Since (cid:96) τ i is a primitive function on the affine hull of the face of F , (cid:96) τ ( p − p ) = (cid:96) τ ( p − p ) = the length of F . We find em τ + m σ α + m σ α = e(cid:96) τ ( p ) + (cid:96) σ ( p ) α + (cid:96) σ ( p ) α = − b(cid:96) τ ( p ) + (cid:96) τ ( p ) + (cid:96) τ ( p ) = (cid:96) τ ( p − p ) . (cid:3) OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 11 Resolution of Newton nondegenerate surface singularities
In this section, we retain the notation introduced in section 3, with the assumption that rk N =3 . We describe Oka’s algorithm which describes explicitly the graph of a resolution of a Newtonnondegenerate Weil divisor of dimension . This algorithm was originally described by Oka [27]for Newton nondegenerate hypersurface singularities in ( C , . The general methods for resolvingNewton nondegenerate hypersurface singularities have been used in e.g. [32] and [3, Chapter 8].6.1. Definition. A canonical subdivision of (cid:52) f is a subdivision (cid:101) (cid:52) f satisfying the following.(i) (cid:101) (cid:52) f is a regular subdivision of (cid:52) f .(ii) If σ ∈ (cid:52) (2) f \ (cid:52) ∗ f , then (cid:101) (cid:52) f | σ is the canonical subdivision (cid:101) (cid:52) σ of (cid:52) σ given in definition 4.2.6.2 . The existence of a canonical subdivision is proved in [27, §3]. We fix such a subdivision (cid:101) (cid:52) f .We will denote by (cid:101) Y the toric variety associated with (cid:101) (cid:52) f . The map (cid:101) Y → Y is denoted by π ,and the strict transform of X under this map is denoted by (cid:101) X . We denote by π X the restriction π | (cid:101) X : (cid:101) X → X . By lemma 2.3, the map (cid:101) Y → Y is proper, hence (cid:101) X → X is proper as well.6.3. Definition.
For i, d ∈ N , define (cid:101) (cid:52) ( i,d ) f = (cid:110) σ ∈ (cid:101) (cid:52) ( i ) f (cid:12)(cid:12)(cid:12) dim( F σ ∩ Γ( f )) = d (cid:111)(cid:101) (cid:52) ∗ ( i,d ) f = (cid:101) (cid:52) ( i,d ) f ∩ (cid:101) (cid:52) ∗ f . Definition.
We start by defining a graph G ∗ as follows. Index the set (cid:101) (cid:52) (1 , f by a set N , i.e.write (cid:101) (cid:52) (1 , f = { σ n | n ∈ N } in such a way that the map N → (cid:101) (cid:52) (1 , f , n (cid:55)→ σ n is bijective. Similarly,index the set (cid:101) ∆ (1 , f ∪ (cid:101) ∆ ∗ (1 , f by N ∗ . Hence N ⊂ N ∗ . The elements of N ∗ are referred to as extendednodes , while N as nodes .Denote by F n the face of Γ + ( f ) corresponding to σ n and by (cid:96) n the primitive integral generatorof σ n . Note that n ∈ N if and only if F n is bounded. For n, n (cid:48) ∈ N ∗ , let t n,n (cid:48) be the length ofthe segment F n ∩ F n (cid:48) if this is a bounded segment of dimension 1. If F n ∩ F n (cid:48) is unbounded, orhas dimension 0, then we set t n,n (cid:48) = 0 . Now, for every pair n and n (cid:48) ∈ N ∗ , we join n, n (cid:48) by t n,n (cid:48) bamboos of type α ( (cid:96) n , (cid:96) n (cid:48) ) /β ( (cid:96) n , (cid:96) n (cid:48) ) , as in fig. 3. This finishes the construction of the graph G ∗ .Denote its set of vertices V ∗ .Define the graph G as the induced full subgraph of G ∗ on the set of vertices V = V ∗ \ ( N ∗ \ N ) . − b − b · · ·− b − b − b − b · · ·· · · − b s − − b s − − b s − − b s − b s − b s n n (cid:48) ... ... Figure 3.
We join n, n (cid:48) ∈ N by t n,n (cid:48) bamboos of the above form, where thesequence b , . . . , b s is defined as b = 1 if α ( (cid:96) n , (cid:96) n (cid:48) ) = 1 , and by a negative continuedfraction expansion α ( (cid:96) n , (cid:96) n (cid:48) ) /β ( (cid:96) n , (cid:96) n (cid:48) ) = [ b , . . . , b s ] otherwise.In order to have a plumbing graph structure on G , we must specify an Euler number and a genusfor each vertex, as well as a sign for each edge. All edges are positive. Vertices appearing on bambooshave genus zero, whereas the genus g n associated with n ∈ N is defined as the number of integralinterior points in the polygon F n .To every extended node n ∈ N ∗ we have associated the cone σ n and its primitive integral generator (cid:96) n . If v , . . . , v s are the vertices appearing on a bamboo, in this order, from n to n (cid:48) ∈ N ∗ , let (cid:96) , (cid:96) , . . . , (cid:96) s +1 be the canonical primitive sequence associated with (cid:96) n , (cid:96) n (cid:48) . We then set (cid:96) v = (cid:96) i for v = v i , i = 1 , . . . , s , and σ v = R ≥ (cid:104) (cid:96) i (cid:105) . This induces a map γ : V → (cid:101) (cid:52) (1) f with the property that γ ( n ) = σ n for n ∈ N ∗ , and (cid:96) v , (cid:96) w generate an element of (cid:101) (cid:52) (2) f if v, w are adjacent in G ∗ .For any v ∈ V , let V v and V ∗ v be the set of neighbours of v in G and G ∗ , respectively. Then thereexists a unique − b v ∈ Z ≤− satisfying − b v (cid:96) v + (cid:88) u ∈V ∗ v (cid:96) u = 0 in N, The number − b v is the Euler number associated with v ∈ V . We note that if v lies on a bamboo, withthe notation of the previous paragraph, v = v i , then − b v = − b i and − b i ≤ − unless α ( (cid:96) n , (cid:96) n (cid:48) ) = 1 .6.5. Remark.
The link of an isolated surface singularity is a rational homology sphere if and onlyif it has a resolution whose graph is a tree and all vertices have genus zero, see e.g. [20]. The aboveconstruction produces such a graph if and only if all integral points on Γ( f ) lie on its boundary ∂ Γ( f ) .Indeed, if P ⊂ Γ( f ) is a vertex which is not on the boundary, then the nodes corresponding tofaces of Γ( f ) containing P lie on an embedded cycle. Similarly, if S ⊂ Γ( f ) is a face of dimension which is not a subset of the boundary, and S contains integral interior points, then the nodescorresponding to the two faces containing S are joined by more than one bamboo, inducing anembedded cycle in G . Finally, if F ⊂ Γ( f ) is a two dimensional face containing interior integralinterior points, then the corresponding node has nonzero genus. The converse is not difficult.The classical case Y = C is discussed in details in [7].6.6. Example.
Let
Σ = R ≥ , and consider standard coordinates x, y, z on Y = C , and the function f ( x, y, z ) = x + x y + y + z . The Newton diagram Γ( f ) consists of two triangular faces, whose intersection is a segment of lengthtwo. The diagram, as well as the graph obtained by Oka’s algorithm can be seen in fig. 4. − ( , , ) ( , , ) − − − − − ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) (0 , , , , − − − , , , , − x x y y z − Figure 4.
A Newton diagram, and the graph G ∗ , with the subgraph G in black.6.7. Proposition.
Let ( X, be a Newton nondegenerate surface singularity. Then the map (cid:101) X → X is a resolution of ( X, whose resolution graph is G .More precisely, (cid:101) X is smooth and the exceptional set E ⊂ (cid:101) X is a normal crossing divisor. For each σ ∈ (cid:101) (cid:52) (1) f , we can enumerate the irreducible components of E σ by γ − ( σ ) so that E σ = (cid:113) v ∈ γ − ( σ ) E v ,where E v is a smooth curve.If γ ( v ) ∈ (cid:101) (cid:52) (1) f \ (cid:101) (cid:52) ∗ , then E v is compact, has genus g v , and its normal bundle in (cid:101) X has Eulernumber − b v . If γ ( v ) ∈ (cid:101) (cid:52) (1) ∗ , then E v is a smooth germ, transverse to a smooth point of theexceptional divisor. OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 13
Furthermore, if v, w ∈ V , then the number of intersection points | E v ∩ E w | equals the number ofedges between v and w in G .Proof. The proof goes exactly as in [27] (cid:3)
Definition.
For v ∈ V ∗ , (recall 3.3 and definition 3.4) let F v = F γ ( v ) , (cid:96) v = (cid:96) γ ( v ) , m v = m γ ( v ) . Lemma.
For v ∈ V , we have − b v (cid:96) v + (cid:88) u ∈V ∗ v (cid:96) u = 0 , − b v m v + (cid:88) u ∈V ∗ v m u = − ( F v ) . Proof.
The first equality follows from construction, see also [27, §6]. The second equality follows from[7, Prop. 4.4.4] and the formula α(cid:96) = β(cid:96) + (cid:96) s +1 , where (cid:96) , (cid:96) , . . . , (cid:96) s +1 is a primitive sequence. (cid:3) Remark. (i) The exceptional divisor E is the union of E σ for which σ ∈ (cid:101) (cid:52) (1) f is a cone whichis not contained in ∂ Σ , or, equivalently, F σ is compact.(ii) If σ ∈ (cid:101) (cid:52) (1 , f , then E σ is a compact smooth irreducible curve. If σ ∈ (cid:101) (cid:52) (1 , f \ (cid:101) (cid:52) ∗ f , then E σ is theunion of t σ disjoint smooth compact rational curves. For σ ∈ (cid:101) (cid:52) ∗ (1 , f , the intersection E σ = V ( σ ) ∩ (cid:101) X is the disjoint union of t smooth curve germs, where t is the length of the segment F σ ∩ Γ( f ) . If σ ∈ (cid:101) (cid:52) (1 , f , then E σ = ∅ (the global divisor D σ does not intersect (cid:101) X ).6.11. Definition.
We denote by L = Z (cid:104) E v | v ∈ V(cid:105) the lattice of integral cycles in (cid:101) X supported onthe exceptional divisor E .6.12. Definition.
Let g ∈ O Y, and denote its restriction by g ∈ O X, . For any v ∈ V ∗ , we define wt v ( g ) = min { (cid:96) v ( p ) | p ∈ supp( g ) } , wt( g ) = (cid:88) v ∈V wt v ( g ) E v ∈ L, wt v ( g ) = max { wt v ( g + h ) | h ∈ I X } , wt( g ) = (cid:88) v ∈V wt v ( g ) E v ∈ L. For σ = γ ( v ) , we also write wt σ instead of wt v , as this is independent of v ∈ γ − ( σ ) .Similarly, for any v ∈ V , let div v be the valuation on O X, associated with the divisor E v , thatis, for g ∈ O X, , denote by div v ( g ) the order of vanishing of the function π ∗ X ( g ) along E v . Set also div( g ) = (cid:88) v ∈V div v ( g ) E v ∈ L. Remark. (i) If σ = γ ( v ) and | γ − ( σ ) | > , then div v is not independent of the choice of v ∈ γ − ( σ ) .(ii) For σ ∈ (cid:101) (cid:52) (1) f , the function wt σ : O Y, \ { } → Z is the valuation on O Y, associated with theirreducible divisor V ( σ ) ⊂ (cid:101) Y , cf. eq. (3.2).(iii) In general, the functions wt v and div v do not coincide on O X, . However, wt v ( g ) ≤ div v ( g ) for any g ∈ O X, and v ∈ V . Furthermore, if p ∈ M and γ ( v ) ∈ (cid:101) (cid:52) (1 ,> f \ (cid:101) (cid:52) ∗ f , then div v ( x p ) =wt v ( x p ) = (cid:96) v ( p ) . In particular, this defines a group homomorphism M → L , p → wt( x p ) .7. The geometric genus
In this section we provide a formula for the delta invariant and geometric genus for an arbitrarygeneralized Newton nondegenerate singularity in terms of its Newton polyhedron. In this section,the rank r of N is under no restriction. Recall that we say that f (or Γ + ( f ) ) is pointed at p ∈ M Q ,if for any σ ∈ (cid:52) (1)Σ we have m σ = (cid:96) σ ( p ) , see definition 3.9.7.1. Remark.
In the proof of theorem 7.3, one of the main steps consists of computing the coho-mology of a line bundle on a toric variety. To do this, we build on classical methods [13, 10]. A moregeneral method to compute such cohomology has been described by Altmann and Ploog in [2].
Definition.
For a point x in an analytic variety X , denote by O X,x the normalization of itslocal ring O X,x . The delta invariant associated with x ∈ X is defined as δ ( X, x ) = dim C O X,x / O X,x . Let (cid:101) X → X be a resolution of the singularity x ∈ X and assume that X has dimension d . Assume,furthermore, that δ ( X, x ) < ∞ , and that the higher direct image sheaves R i π ∗ O (cid:101) X , i > , areconcentrated at x . The geometric genus p g = p g ( X, is defined as ( − d − p g ( X, x ) = δ ( X, x ) + d − (cid:88) i =1 ( − i h i ( (cid:101) X, O (cid:101) X ) . We say that ( X, x ) is rational if δ ( X, x ) = 0 and h i ( (cid:101) X, O (cid:101) X ) = 0 for i > .7.3. Theorem.
Let ( X, ⊂ ( Y, be a Newton nondegenerate Weil divisor of dimension d = r − .(i) We have the following canonical identifications O X, / O X, ∼ = (cid:77) p ∈ M (cid:101) H (Γ + ( x p f ) \ Σ ∨ , C ) ,H i ( (cid:101) X, O (cid:101) X ) ∼ = (cid:77) p ∈ M (cid:101) H i (Γ + ( x p f ) \ Σ ∨ , C ) , i > . In particular, if these vector spaces have finite dimension, then δ ( X,
0) = (cid:88) p ∈ M (cid:101) h (Γ + ( x p f ) \ Σ ∨ , C ) ,p g ( X,
0) = ( − d − (cid:88) p ∈ M (cid:101) χ (Γ + ( x p f ) \ Σ ∨ , C ) , where (cid:101) χ denotes the reduced Euler characteristic , that is, the alternating sum of ranks ofreduced singular cohomology groups.(ii) We have (cid:101) h d − (Γ + ( x p f ) \ Σ ∨ , C ) = (cid:40) if ∈ Γ ∗ + ( x p f ) ◦ \ Γ + ( x p f ) ◦ , else . In particular, h d − ( (cid:101) X, O (cid:101) X ) = | M ∩ Γ ∗ + ( f ) ◦ \ Γ + ( f ) ◦ | (recall definition 3.5).(iii) Assume that f is Q -pointed, that d ≥ , and that ( X, has only rational singularitiesoutside the origin. Then ( X, is normal and h i ( (cid:101) X, O (cid:101) X ) = 0 for ≤ i < d − . Corollary.
Assume that d = 2 and ( X, is normal. Then p g ( X,
0) = | M ∩ Γ ∗ + ( f ) ◦ \ Γ + ( f ) ◦ | . (cid:3) This generalizes a result of Merle and Teissier [19] valid for the classical case
Σ = R ≥ .7.5. Corollary.
Assume that d = 1 and ( X, is an irreducible germ of a curve, and that σ ∈ (cid:101) (cid:52) (1) f satisfies F σ = Γ( f ) (cf. proposition 4.5(i)). Then δ ( X, is the number of unordered pairs (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ∈ Σ ◦ ∩ N satisfying (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) = (cid:96) σ .Proof. Let P ( (cid:96) σ ) be the parallelogram introduced in the proof of proposition 4.5. The diagonalsplits P ( (cid:96) σ ) into two triangles, T and T , say. If (cid:96) (cid:48) ∈ T ◦ , then (cid:96) σ − (cid:96) (cid:48) ∈ T ◦ . This induces a bijectionbetween elements (cid:96) (cid:48) ∈ T ◦ ∩ N and unordered pairs { (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) } ⊂ Σ ◦ ∩ N adding up to (cid:96) σ . By rotatingby π/ as in the proof of proposition 4.5, T ◦ ∩ N is in bijection with M ∩ Γ ∗ + ( f ) ◦ \ Γ + ( f ) ◦ . (cid:3) Remark.
Assume that d ≥ , and that X is rational outside { } . Then, for < i < d − , wehave H i ( (cid:101) X, O (cid:101) X ) ∼ = H i ( (cid:101) X \ E, O (cid:101) X ) ∼ = H i ( X \ { } , O X ) ∼ = H i +1 { } ( X, O X ) . Here, the first isomorphism comes from the long exact sequence for cohomology with support in E , and the vanishing H iE ( (cid:101) X, O (cid:101) X ) = 0 , for i < d [14, Corollary 3.3]. The second isomorphismfollows from the rationality assumption, and the Leray spectral sequence. The third isomorphism OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 15 comes from the similar long exact sequence for cohomology with support in { } , and the fact that H j ( X, O X ) = 0 for j > , if we choose a Stein representative X of the germ ( X, . This last longexact sequence furthermore gives H { } ( X, O X ) ∼ = H ( X \ { } , O X ) H ( X, O X ) ∼ = O X, / O X, . Therefore, in this case, the groups described in theorem 7.3 are closely related with the depth of O X, . In particular, the conclusion of theorem 7.3(iii) is that ( X, is a Cohen–Macaulay ring.If f is pointed at p ∈ M , then this statement can be proved as follows. Since ( X, is a Cartierdivisor in ( Y, , cf. proposition 3.10(ii), and ( Y, is Cohen–Macaulay [9, Theorem 6.3.5] so is ( X, [9, Theorem 2.1.3]. Proof of theorem 7.3.
To prove (i), we use results and notation from [10, §7], see also [13, 3.5]. Define D m = (cid:88) (cid:110) m σ D σ (cid:12)(cid:12)(cid:12) σ ∈ (cid:101) (cid:52) (1) f (cid:111) . Then D m + (cid:101) X is the divisor of the pullback of f to (cid:101) Y , and we have a short exact sequence → O (cid:101) Y ( D m ) · f → O (cid:101) Y → O (cid:101) X → . By [10, Corollary 7.4], we have H i ( (cid:101) Y , O (cid:101) Y ) = 0 for all i > . Furthermore, H ( (cid:101) X, O (cid:101) X ) ∼ = O X, , andthe image of H ( (cid:101) Y , O (cid:101) Y ) = O Y, in O X, is O X, . Therefore, O X, / O X, ∼ = H ( (cid:101) Y , O (cid:101) Y ( D m )) , and H i ( (cid:101) X, O (cid:101) X ) ∼ = H i +1 ( (cid:101) Y , O (cid:101) Y ( D m )) , i > . Denote by g the order function defined in [10, §6] (using the natural trivialization of O (cid:101) Y ( D m ) onthe open torus) g : | (cid:101) (cid:52) f | → R , g ( (cid:96) ) = − min { (cid:96) ( q ) | q ∈ Γ + ( f ) } and define the sets Z p = (cid:110) (cid:96) ∈ | (cid:101) (cid:52) f | (cid:12)(cid:12)(cid:12) (cid:96) ( p ) ≥ g ( (cid:96) ) (cid:111) , p ∈ M. We note that Z p is a convex cone and that ∈ Z p for all p ∈ M . By [10, Theorem 7.2], we haveisomorphisms H i +1 ( (cid:101) Y , O (cid:101) Y ( D m )) ∼ = (cid:77) p ∈ M H i +1 Z p ( | (cid:101) (cid:52) f | , C ) . Since | (cid:101) (cid:52) f | = Σ is a convex set, the long exact sequence associated with cohomology with supportsprovides, for any p ∈ M ∼ = (cid:101) H i ( | (cid:101) (cid:52) f | , C ) → (cid:101) H i ( | (cid:101) (cid:52) f | \ Z p , C ) ∼ = H i +1 Z p ( | (cid:101) (cid:52) f | , C ) → (cid:101) H i +1 ( | (cid:101) (cid:52) f | , C ) ∼ = 0 . To finish the proof of (i), we will show that for any p ∈ M , the spaces | (cid:101) (cid:52) f | \ Z p = Σ \ Z p and Γ + ( x p f ) \ Σ ∨ are in fact homotopically equivalent. We start by noting that the the condition Z p ⊂ ∂ | (cid:101) (cid:52) f | (including the case when Z p = ∅ ) is equivalent to ∈ Γ + ( x p f ) \ Γ( x p f ) . If this happensthen we can choose a q ∈ Σ ∨ small so that − q ∈ Γ + ( x p f ) \ Γ( x p f ) as well, and so Γ + ( x p f ) \ Σ ∨ isstar-shaped with center − q . In particular, in this case, Σ \ Z p ∼ { a point } ∼ Γ + ( x p f ) \ Σ ∨ , where ∼ denotes the homotopy equivalence. Thus, in what follows, we assume that Z p contains aninterior point in Σ , equivalently, / ∈ Γ + ( x p f ) \ Γ( x p f ) .Choose (cid:96) ∈ Σ ◦ and q ∈ (Σ ∨ ) ◦ satisfying (cid:96) ( q ) = 1 and define the hyperplanes H = { (cid:96) ∈ N R | (cid:96) ( q ) = 1 } , H ∨ = { q ∈ M R | (cid:96) ( q ) = 1 } . Then, seeing H and H ∨ as linear spaces by choosing origins (cid:96) , q , the pairing H × H ∨ (cid:51) ( (cid:96), q ) (cid:55)→ (cid:96) ( q ) − is nondegenerate and the polyhedrons H ∩ Σ and H ∨ ∩ Σ ∨ are each others polar sets as in[13, 1.5].Since ∈ Z p , we have Σ \ Z p ∼ ( H ∩ Σ \ Z p ) × R ∼ H ∩ Σ \ Z p . By the assumptions made above, there is an (cid:96) ∈ Z p ∩ Σ ◦ . Both Σ ∩ H and Z p ∩ H are compact convexpolyhedrons in H . Projection away from (cid:96) onto ∂ ( H ∩ Σ) then induces a homotopy equivalence H ∩ Σ \ Z p ∼ H ∩ ∂ Σ \ Z p . By projection, we mean that any element in a ray r = (cid:96) + R > (cid:96) (cid:48) ⊂ H maps to the unique elementin r ∩ ∂ ( H ∩ Σ) . By lemma 7.7(i), this has the subset ∪ (cid:8) H ∩ σ (cid:12)(cid:12) σ ∈ (cid:52) ∗ f , H ∩ σ ∩ Z p = ∅ (cid:9) as a strong deformation retract. All this yields(7.1) Σ \ Z p ∼ ∪ (cid:8) H ∩ σ (cid:12)(cid:12) σ ∈ (cid:52) ∗ f , σ ∩ Z p = { } (cid:9) . Using a projection, this time onto ∂ Σ ∨ in M , having as center any element in (Σ ∨ ∩ Γ + ( x p f )) ◦ ,we get a homotopy equivalence Γ + ( x p f ) \ Σ ∨ ∼ Γ + ( x p f ) ◦ ∩ ∂ Σ ∨ . By lemma 7.7(ii), we have a homotopy equivalence Γ + ( x p f ) ◦ ∩ ∂ Σ ∨ ∼ ∪ (cid:8) ( σ ⊥ ∩ Σ ∨ ) ◦ (cid:12)(cid:12) σ ∈ (cid:52) Σ , σ (cid:54) = { } , ( σ ⊥ ∩ Σ ∨ ) ◦ ∩ Γ + ( x p f ) ◦ (cid:54) = ∅ (cid:9) . Since, by assumption made above, / ∈ Γ + ( x p f ) ◦ , and so the right hand side above has a free actionby R > which has a section given by intersection with H ∨ . Furthermore, one checks that if σ ∈ (cid:52) ∗ f ,then ( σ ⊥ ∩ Σ ∨ ) ◦ ∩ Γ + ( x p f ) ◦ (cid:54) = ∅ ⇔ ∀ (cid:96) ∈ σ \ { } : (cid:96) ( p ) + m (cid:96) < . Here, the condition on the left is equivalent to σ ∩ Z p = { } , so(7.2) Γ + ( x p f ) \ Σ ∨ ∼ ∪ (cid:8) H ∩ ( σ ⊥ ∩ Σ ∨ ) ◦ (cid:12)(cid:12) σ ∈ (cid:52) ∗ f , σ ∩ Z p = { } (cid:9) . Now, consider the CW structure K given by the cells H ∩ σ in H ∩ ∂ Σ and K (cid:48) given by cells H ∨ ∩ ( σ ⊥ ∩ Σ ∨ ) in H ∨ ∩ ∂ Σ ∨ . Using barycentric subdivision, one obtains a homeomorphism φ : H ∩ ∂ Σ → H ∨ ∩ ∂ Σ ∨ , sending the center of a cell H ∩ σ to the center of the dual cell H ∩ σ ∨ , thusidentifying K with the dual of K (cid:48) . By this identification, the left hand side of eq. (7.2) is a regularneighbourhood around the image under φ of the left hand side of eq. (7.1). This concludes (i).Next, we prove (ii). By the above discussion, the result is clear in the cases when Z p = ∅ or Z p ⊂ ∂ Σ . Assuming that this is not the case, the complex, say, A , on the right hand side of eq. (7.1)is a closed subset of H ∩ ∂ Σ ∼ S d − . Then h d − ( A, C ) = 0 , unless A = H ∩ ∂ Σ , in which case h d − ( A, C ) = 1 . But this is equivalent to (cid:96) ( p ) + m (cid:96) < for all (cid:96) ∈ ∂ Σ \ , that is, ∈ Γ ∗ + ( x p f ) ◦ .For (iii), we will show that Γ + ( x p f ) \ Σ ∨ has trivial homology in degrees i < d − for all p ∈ M .By assumption, there is a q ∈ M Q so that for σ ∈ (cid:52) (1)Σ we have m σ = (cid:96) σ ( q ) . We can again assumethat ∈ Γ + ( x p f ) \ Γ( x p f ) . We must show that (cid:101) h i ( A, C ) = 0 for i < d − , where A is the righthand side of eq. (7.1). We note that by definition, A consists of cells H ∩ σ for σ ∈ (cid:52) ∗ f satisfying ∀ (cid:96) ∈ H ∩ σ : (cid:96) ( p ) < − m (cid:96) . Define similarly A Σ = ∪ { H ∩ σ | σ ∈ (cid:52) ∗ Σ , ∀ (cid:96) ∈ H ∩ σ : (cid:96) ( p ) < − (cid:96) ( q ) } . Define A q = { (cid:96) ∈ H ∩ ∂ Σ | (cid:96) ( p ) < − (cid:96) ( q ) } . This space can be either S d − , an d − dimensional ball, or empty. In each case, (cid:101) H i ( A q , C ) = 0 for i < d − . We will show that A q ⊃ A Σ ⊂ A , and that these inclusions are homotopy equivalences.For the first one, in fact, this is clear by definition and lemma 7.7(i).For the second one, denote by A i Σ the i -skeleton of the complex A Σ , and define similarly A i = A \ ∪ (cid:110) σ ◦ (cid:12)(cid:12)(cid:12) σ ∈ (cid:52) ∗ ( ≥ i +2)Σ (cid:111) . We will prove by induction on i that A i Σ ⊂ A i and that this is a homotopy equivalence. The case i = 0 follows from the pointed condition: assuming σ ∈ (cid:52) ∗ (1)Σ is a ray, there is a t > so that H ∩ σ = { t(cid:96) σ } . By assumption, we have m (cid:96) σ = (cid:96) σ ( q ) , so that H ∩ σ ⊂ A Σ if and only if H ∩ σ ⊂ A .Since A consists only of such zero-cells, we get A = A . OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 17
Next, assume that for some i > we have an inclusion A i − ⊂ A i − which is a homotopyequivalence. Let σ ∈ (cid:52) ∗ ( i +1)Σ provide an i -cell H ∩ σ in A Σ . In this case, we want to show that H ∩ σ ⊂ A i . In fact, we have ∂ ( H ∩ σ ) ⊂ A i − , hence ∂ ( H ∩ σ ) ⊂ A i − , by induction. But bythe rationality assumption on the transverse type, it follows from (ii) and lemma 3.12 that we musthave σ ⊂ A i , thus A i Σ ⊂ A i .To show that this inclusion is a homotopy equivalence, let σ ∈ (cid:52) ∗ ( i +1) f provide an i -cell H ∩ σ which is not in A i Σ . By definition, we see that σ (cid:54)⊂ A i as well. In fact, similarly as in the proof of (i),the inclusion ∂ ( H ∩ σ ) ∩ A i ⊂ ( H ∩ σ ) ∩ A i is a strong deformation retract. Since these cells, alongwith A i Σ provide a finite closed covering, these glue together to form a strong deformation retract A i → A i Σ . (cid:3) Lemma.
Let
K, L ⊂ R N . Assume that K is given as a finite disjoint union K = ∪ α ∈ I K α ofrelatively open convex polyhedrons K α , i.e. each K α is given by a finite number of affine equationsand strict inequalities. Furthermore, assume the following two conditions: (cid:96) If F is the face of K α for some α , then F = K β for some β . (cid:96) For any α, β , the intersection K α ∩ K β is a face of both K α and K β .Note that the polyhedrons K α may be unbounded. In this case(i) Assume that K is compact and L is convex. Then the inclusion (7.3) (cid:91) α ∈ I (cid:8) K α (cid:12)(cid:12) K α ∩ L = ∅ (cid:9) ⊂ K \ L is a strong deformation retract.(ii) Assume that L is convex. Then the inclusion (cid:91) α ∈ I { K α | K α ∩ L (cid:54) = ∅} ⊂ K ∩ L is a strong deformation retract.Proof. We prove (i), similar arguments work for (ii). We use induction on the number of α with K α ∩ L (cid:54) = ∅ . Indeed, if this number is zero, then the inclusion in eq. (7.3) is an equality.Otherwise, there is an α with K α ∩ L (cid:54) = ∅ . Define I (cid:48) = (cid:8) α ∈ I (cid:12)(cid:12) K α (cid:54)⊃ K α (cid:9) (cid:40) I, K (cid:48) = ∪ α ∈ I (cid:48) K α . Then the left hand side of eq. (7.3) does not change if we replace I by I (cid:48) . Therefore, using theinduction hypothesis, it is enough to show that the inclusion K (cid:48) \ L ⊂ K \ L is a homotopy equivalence.We do this by constructing a deformation retract h : K \ L × [0 , → K \ L . For this, we use thefinite closed covering K α \ L , α ∈ I of K \ L . It is then enough to define the restriction h α of h to ( K α \ L ) × [0 , for α ∈ I in such a way that these definitions coincide on intersections.For any α ∈ I (cid:48) , we define h α ( x, t ) = x . Let q ∈ K α ∩ L . If α ∈ I \ I (cid:48) , then q ∈ K α , and wedefine h α by projecting away from q , that is, for any x ∈ K α there is a unique y in the intersectionof ∂K α \ K α and they ray starting at q passing through x . We define h α ( x, t ) = (1 − t ) x + ty .One readily verifies that these functions are continuous, agree on intersections of their domains anddefine a strong deformation retract. (cid:3) Canonical divisors and cycle
In this section we describe possible canonical divisors for (cid:101) Y = Y (cid:101) (cid:52) f and (cid:101) X . Furthermore, in thecase d = 2 , we give a formula for the canonical cycle.8.1. Definition.
Let (cid:101) X → X be a resolution of singularities of an ( r − -dimensional singularity.A canonical divisor K (cid:101) X on (cid:101) X is any divisor satisfying O (cid:101) X ( K (cid:101) X ) ∼ = Ω r − (cid:101) X .If r = 3 then let E = ∪ v ∈V E v be the exceptional divisor of a resolution (cid:101) X → X , where E v arethe irreducible components of E . Recall that we denoted by L the lattice of integral cycles in (cid:101) X supported on the exceptional divisor E : that is, L = Z (cid:104) E v | v ∈ V(cid:105) . We also set L Q = L ⊗ Q and L (cid:48) = Hom( L, Z ) ∼ = { l (cid:48) ∈ L Q | ∀ l ∈ L : ( l (cid:48) , l ) ∈ Z } , where ( · , · ) denotes the intersection form, extended linearly to L Q . Moreover, set E ∗ v ∈ L (cid:48) for theunique rational cycle satisfying ( E v , E ∗ v ) = − and ( E w , E ∗ v ) = 0 for w (cid:54) = v .In this surface singularity case the canonical cycle Z K ∈ L (cid:48) is the unique rational cycle on (cid:101) X supported on the exceptional divisor, satisfying the adjunction formula ( E v , Z K ) = − b v + 2 − g v for any irreducible component E v of the exceptional divisor, where − b v is the Euler number of thenormal bundle of E v ⊂ (cid:101) X , and g v is the genus of E v (we assume here that the components E v ofthe exceptional divisor are smooth).8.2. Remark.
The cycles Z K and E ∗ v are well defined, since the intersection matrix, with entries ( E v , E w ) , associated with any resolution is negative definite. Notice also that any two canonicaldivisors are linearly equivalent, and that any canonical divisor K is numerically equivalent to − Z K .However, it can happen that O (cid:101) X ( K (cid:101) X + Z K ) has infinite order in the Picard group.8.3. Proposition.
Fix any r . Let ( X, ⊂ ( Y, be a Newton nondegenerate Weil divisor, and (cid:101) (cid:52) f a subdivision of the normal fan (cid:52) f so that (cid:101) Y → Y is an embedded resolution. Then the divisors (8.1) K (cid:101) Y = − (cid:88) σ ∈ (cid:101) (cid:52) (1) f D σ ∈ Div( (cid:101) Y ) , K (cid:101) X = − (cid:88) σ ∈ (cid:101) (cid:52) (1) f (1 + m σ ) E σ ∈ Div( (cid:101) X ) are possible canonical divisors for (cid:101) Y and (cid:101) X , respectively.Furthermore, in the surface case ( r = 3 ), the canonical cycle on (cid:101) X is given by the formula (8.2) Z K − E = wt( f ) − (cid:88) ( m n + 1) E ∗ v , where the sum to the right runs through edges { n, v } in the graph G ∗ so that n ∈ N ∗ \ N and v ∈ V (and the identity is in L ).Proof. For K (cid:101) Y , see e.g. 4.3 of [13]. Since the divisor (cid:101) X + (cid:80) σ ∈ (cid:101) (cid:52) (1) f m σ D σ = ( π ∗ f ) is principal in (cid:101) Y (and D σ | (cid:101) X = E σ ), the adjunction formula gives K (cid:101) X = (cid:16) K (cid:101) Y + (cid:101) X (cid:17)(cid:12)(cid:12)(cid:12) (cid:101) X = − (cid:88) σ ∈ (cid:101) (cid:52) (1) f ( m σ + 1) E σ , which proves eq. (8.1). To prove eq. (8.2), it is enough to show that in L for all v ∈ V ,(8.3) ( Z K − E, E v ) = (cid:16) wt( f ) − (cid:88) ( m n + 1) E ∗ v , E v (cid:17) , where the sum is as in eq. (8.2). Recall that wt( f ) = (cid:80) v ∈V m v E v . We note that the adjunctionformula gives ( Z K − E, E v ) = 2 − g v − δ v for all v ∈ V , where δ v is the valency of the vertex v in G , and g v is the genus of E v . Furthermore, it follows from definition 6.4 that if v ∈ V , then (cid:96) δ v = 1 if and only v is on the end of a bamboo joining a node n ∈ N and an extended node n (cid:48) ∈ N ∗ \ N . In this case, v has exactly one neighbour in V ∗ \ V in the graph G ∗ . (cid:96) δ v = 2 if and only v is on a bamboo joining two extended nodes, and is not of the formdescribed in the previous item. (cid:96) δ v ≥ if and only if v is a node.Consider first the case δ v = 1 , and let n be the unique neighbour of v in N ∗ \ N . It follows fromlemma 6.9 that (wt( f ) , E v ) = − m n , since F v is a segment, and so has area zero. As a result, theright hand side of eq. (8.3) is Z K − E, E v ) .Next, assume that δ v = 2 . Then both sides of eq. (8.3) vanish (use again lemma 6.9).Assume finally that v ∈ N . Then, v has no neighbours in N ∗ \ N . Furthermore, δ v coincideswith the number of integral points on the boundary of F v , since each edge adjacent to v can be seento correspond to a primitive segment of the boundary. By using Pick’s theorem and lemma 6.9, wetherefore get ( Z K − E, E v ) = 2 − g v − δ v = − ( F v ) = ( E v , wt( f )) , which finishes the proof. (cid:3) OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 19
Remark. As m σ depends on the choice of f up to a x p multiplication, the right hand side of thesecond formula from eq. (8.1) depends on this choice too. In fact, the monomial rational function x p realizes the linear equivalence between the two divisors K (cid:101) X associated with two such choices.9. Gorenstein surface singularities
In this section we prove theorem 9.6, which characterizes nondegenerate normal surface Gorensteinsingularities by their Newton polyhedron. The key technical lemmas 9.10 and 9.11 provide the toolsfor the proof. They are proved using vanishing of certain cohomology groups calculated by toricmethods. In the first lemma, the restriction r = 3 is not needed. However, the second lemma relieson the negative definiteness of the intersection form, restricting our result to the surface case.9.1. Definition.
Let f and (cid:52) f be as above. We say that Γ + ( f ) , or f , is ( Q -) Gorenstein-pointed ifthere exists a p ∈ M ( p ∈ M Q ) such that (cid:96) σ ( p ) = m σ + 1 for all σ ∈ (cid:52) ∗ (1 , f .9.2. Example.
Recall that ( Y, is Gorenstein if and only if there is a p ∈ M satisfying (cid:96) σ ( p ) = 1 for all σ ∈ (cid:52) (1)Σ , see e.g. [8], Theorem 6.32. Therefore, if ( X, is Cartier, and (cid:52) ∗ f = (cid:52) ∗ Σ , then f is Gorenstein pointed (since m σ = 0 for σ ∈ (cid:52) (1)Σ ). Furthermore, ( X, is Gorenstein since ( Y, isGorenstein and f forms a regular sequence.Similarly, ( Y, is Q -Gorenstein if there is a p ∈ M Q satisfying (cid:96) σ ( p ) = 1 for all σ ∈ (cid:52) (1)Σ , see e.g.[1]. Therefore, if ( X, is Cartier, and (cid:52) ∗ f = (cid:52) ∗ Σ , then f is Q -Gorenstein pointed.9.3. Remark.
Though the two combinatorial conditions in definitions 3.9 and 9.1 look very simi-lar, they codify two rather different geometrical properties. Being ‘pointed’ codifies an embeddingproperty, namely that ( X, ⊂ ( Y, is Cartier, see proposition 3.10. However, being ‘Gorensteinpointed’ codifies an abstract property of the germ ( X, , namely its Gorenstein property, see theo-rem 9.6 below.9.4 . Recall also that ( X, is Gorenstein if it admits a Gorenstein form. A Gorenstein form is anowhere vanishing section in H ( X \ , Ω X \ ) = H ( (cid:101) X \ E, Ω (cid:101) X \ E ) . A Gorenstein pluri-form is anowhere vanishing section in H ( (cid:101) X \ E, (Ω (cid:101) X \ E ) ⊗ k ) for some k ∈ Z > .In this section K (cid:101) Y and K (cid:101) X are canonical divisors with a choice as in eq. (8.1).9.5. Definition.
Let ω f be some meromorphic 2-form on (cid:101) X whose divisor ( ω f ) is K (cid:101) X .9.6. Theorem.
Assume that ( X, ⊂ ( Y, is a normal Newton nondegenerate surface singularity(i.e. r = 3 ). The following conditions are equivalent:(i) f is Gorenstein-pointed at some p ∈ M .(ii) There exists a p ∈ M so that for all v ∈ V ∗ \ V we have (cid:96) v ( p ) = m v + 1 .(iii) There exists a p ∈ M so that for all v ∈ V we have (cid:96) v ( p ) = m v + 1 − m v ( Z K ) .(iv) There exists a p ∈ M so that x p ω f is a Gorenstein form.(v) ( X, is Gorenstein.When these conditions hold, (i), (ii), (iii) and (iv) uniquely identify the same point p . In fact, the analogues of parts (i)–(iv) are equivalent over rational points p ∈ M Q as well.9.7. Proposition.
Under the assumption of theorem 9.6, the following conditions are equivalent,and imply that ( X, is Q -Gorenstein:(i) f is Q -Gorenstein-pointed at some p ∈ M Q .(ii) There exists a p ∈ M Q so that for all v ∈ V ∗ \ V we have (cid:96) v ( p ) = m v + 1 .(iii) There exists a p ∈ M Q so that for all v ∈ V we have (cid:96) v ( p ) = m v + 1 − m v ( Z K ) .(iv) There exists a p ∈ M Q so that x kp ( ω f ) ⊗ k is a Gorenstein pluri-form for some k ∈ Z > .Furthermore, all these these conditions identify the very same p uniquely.Proof. (ii) is a rephrasing of (i), since (cid:52) ∗ (1 , f = V ∗ \ V . (ii) ⇒ (iii) For any p ∈ M Q consider the cycles Z := (cid:88) v ∈V (cid:96) v ( p ) E v ∈ L Q , Z := (cid:88) ( m n + 1) E ∗ v ∈ L Q , where the sum runs over edges { n, v } in G ∗ so that n ∈ V ∗ and v ∈ V (as in eq. (8.2)), and Z ∗ := (cid:80) n ∈V ∗ \V (cid:96) n ( p ) E n (where all these E n ’s are the noncompact curves in (cid:101) X ).If { n, v } is an edge as above, then ( Z , E v ) = − ( m n +1) . Moreover, ( Z ∗ , E v ) (cid:101) X = (cid:96) n ( p ) . Therefore,by assumption (ii), ( Z ∗ + Z , E u ) (cid:101) X = 0 for any u ∈ V . On the other hand, by lemma 6.9, ( Z ∗ + Z , E u ) (cid:101) X = 0 for any u ∈ V as well. Hence Z = Z . But by eq. (8.2) m u ( Z ) = m v + 1 − m v ( Z K ) .(iii) ⇒ (ii) With the above notations, (iii) shows that Z = Z . Let { n, v } be an edge as above, let w ∈ V be the other neighbour of v , and note that E v = b v E ∗ v − E ∗ w in L (cid:48) . Then, m n + 1 = ( Z , − E v ) = ( Z , − b v E ∗ v + E ∗ w ) = (cid:96) v ( p ) b v − (cid:96) w ( p ) = (cid:96) n ( p ) (in the last equality use lemma 6.9).For (ii) ⇔ (iv) use the second identity of eq. (8.1). (cid:3) Remark.
Similarly as in theorem 9.6, one may ask whether the equivalent cases in 9.7 areequivalent with the property that ( X, is Q -Gorenstein. If f is Q -Gorenstein-pointed at p ∈ M Q ,then (iv) implies that ( X, is Q -Gorenstein. The converse does not hold , as seen by the followingexample.Let N = Z and Σ = R ≥ (cid:104) (1 , , , (0 , , , (1 , , , (0 , , (cid:105) , f ( x ) = x (0 , , + x (1 , , + x (0 , , + 2 x (1 , , − . Write σ i , i = 1 , , , for the rays generated by the vector specified above and denote by m i thecorresponding multiplicities. We find m = m = m = 0 and m = 1 . As a result, since the linearequation · p = has no solution, hence f is not Q -Gorenstein pointed. (0 , , , , , ,
1) (1 , , −
1) (1 , , , , , ,
1) (1 , , , , − G ∗ G Γ( f ) Figure 5.
A Newton diagram, and the output of Oka’s algorithm. The dottedline shows the intersection of the affine hull of the only face of the diagram inter-sected with ∂ Σ ∨ . For simplicity, here in G ∗ we have blown down the ( − -verticesconstructed in the last paragraph of 4.1.On the other hand, one verifies that the Weil divisor defined by f is normal using theorem 7.3.Furthermore, Oka’s algorithm shows that this singularity has a resolution with an exceptional divisorconsisting of a single rational curve with Euler number − . Such a singularity is a cyclic quotientsingularity. In particular, it is Q -Gorenstein. OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 21 . Next, we focus on the proof of theorem 9.6. The equivalences of the first four cases followfrom (or, as) proposition 9.7. For (i) ⇒ (v) note that if f is Gorenstein-pointed at p ∈ M then x p ω f trivializes the canonical bundle. The implication (v) ⇒ (i) will be proved below based on two lemmas.9.10. Lemma.
Let g ∈ H ( (cid:101) X \ E, O (cid:101) X ( K (cid:101) X )) , that is, g is a meromorphic function on the complementof the exceptional divisor in (cid:101) X satisfying (9.1) ( g ) ≥ − K (cid:101) X | (cid:101) X \ E = (cid:88) v ∈V ∗ \V ( m v + 1) E v . Then, there exists a Laurent series g ∈ O Y, [ x M ] satisfying ( π ∗ g ) | (cid:101) X \ E = g and (9.2) ∀ σ ∈ (cid:101) (cid:52) ∗ (1) f : wt σ g ≥ m σ + 1 . Proof.
Let I = H ( (cid:101) X \ E, O (cid:101) X ( K (cid:101) X )) and let J be the set of meromorphic functions obtained as arestriction of Laurent series satisfying eq. (9.2). We want to show that I = J .We immediately see J ⊂ I . In fact, this inclusion fits into an exact sequence as follows. Recallthe notation D m = (cid:80) σ ∈ (cid:101) (cid:52) (1) f m σ D σ from the proof of theorem 7.3, and K (cid:101) Y = − (cid:80) σ ∈ (cid:101) (cid:52) (1) f D σ . Also,define D c as the union of compact divisors in (cid:101) Y , that is, ∪ σ D σ for σ (cid:54)∈ (cid:101) (cid:52) ∗ (1) f . Since ( π ∗ f ) = (cid:101) X + D m ,we have a short exact sequence of sheaves → O (cid:101) Y \ D c ( K (cid:101) Y ) · f → O (cid:101) Y \ D c ( − D m + K (cid:101) Y ) → O (cid:101) X \ E ( − D m + K (cid:101) Y ) → yielding a long exact sequence of cohomology groups. We have I = H ( (cid:101) X \ E, O (cid:101) X \ E ( − D m + K (cid:101) Y )) , since K (cid:101) X = ( − D m + K (cid:101) Y ) | (cid:101) X . Furthermore, since (cid:101) Y is normal, H ( (cid:101) Y \ D c , O (cid:101) Y \ D c ( − D m + K (cid:101) Y )) isthe set of Laurent series satisfying eq. (9.2). Thus, its image in I is J . Therefore, the quotient I/J injects into H ( (cid:101) Y \ D c , O (cid:101) Y \ D c ( K (cid:101) Y )) . On the other hand,(9.3) H ( (cid:101) Y \ D c , O (cid:101) Y \ D c ( K (cid:101) Y )) ∼ = (cid:77) p ∈ M H Z ( p ) ( ∂ Σ , C ) , where, following Fulton [13], ψ K : ∂ Σ → R is the unique function restricting to linear function onall σ ∈ (cid:101) (cid:52) ∗ f , and satisfying ψ K ( (cid:96) σ ) = 1 for σ ∈ (cid:101) (cid:52) (1) ∗ f , and for p ∈ M we set Z ( p ) = { (cid:96) ∈ ∂ Σ | (cid:96) ( p ) ≥ ψ K ( (cid:96) ) } . Firstly, since ∂ Σ is contractible, we find H Z ( p ) ( ∂ Σ , C ) ∼ = (cid:101) H ( ∂ Σ \ Z ( p ) , C ) . Secondly, define Z (cid:48) ( p ) as the union of those cones σ ∈ (cid:101) (cid:52) ∗ f satisfying p | σ ≥ (i.e. (cid:96) ( p ) ≥ for all (cid:96) ∈ σ ), and let Z (cid:48)(cid:48) ( p ) be the set of (cid:96) ∈ ∂ Σ satisfying (cid:96) ( p ) ≥ . By lemma 7.7, the inclusions ∂ Σ \ Z ( p ) ⊂ ∂ Σ \ Z (cid:48) ( p ) ⊃ ∂ Σ \ Z (cid:48)(cid:48) ( p ) are strong deformation retracts. But the right hand side above is either a contractible set, or it hasthe homotopy of S r − . In particular, it is connected, by our assumption r > , and so eq. (9.3)vanishes. (cid:3) Lemma.
Assume that ( X, is a Gorenstein normal surface singularity, i.e. r = 3 , and thatwe have a Gorenstein form ω on (cid:101) X \ E . Thus, − K (cid:101) X − Z K is linearly trivial, and there exists g ∈ H ( (cid:101) X, O (cid:101) X ( K (cid:101) X + Z K )) , ( g ) = ( ω ) − ( ω f ) = − Z K − K (cid:101) X . Then there is a g ∈ O Y, [ x M ] satisfying (9.4) ( π ∗ g ) | (cid:101) X = g and ∀ v ∈ V ∗ : wt v ( g ) = div v ( g ) . Proof.
By the previous lemma 9.10, we can find a g satisfying g | (cid:101) X \ E = g and eq. (9.2). Let A = ( g ) and B = (cid:80) v ∈V ∗ wt v ( g ) E v . We want to prove that A = B . Both A and B are supported in theexceptional divisor and the noncompact curves E v for v ∈ V ∗ \ V , and by our assumptions, theyhave the same multiplicity along this noncompact part. Thus, A − B is supported on the exceptionaldivisor. Furthermore, we have wt v ( g ) ≤ div v ( g ) for v ∈ V , thus B − A ≤ .For the reverse inequality, note first that ( A, E v ) = 0 for all v ∈ V since A is principal. Forany v ∈ V , let q ∈ M be an element of the support of the principal part of g with respect to (cid:96) v ,i.e. q ∈ supp( g ) and (cid:96) v ( q ) = wt v ( g ) . By definition, we also have (cid:96) u ( q ) ≥ wt u ( g ) for all u ∈ V ∗ v .Therefore, ( B, E v ) = − b v wt v ( g ) + (cid:88) { wt u ( g ) | u ∈ V ∗ v } ≤ − b v (cid:96) v ( q ) + (cid:88) { (cid:96) u ( q ) | u ∈ V ∗ v } = 0 . As a result, B − A is in the Lipman cone, and so, B − A ≥ , proving eq. (9.4). (cid:3) Proof of theorem 9.6.
The first four conditions are equivalent by proposition 9.7, and (iv) clearlyimplies (v).Assuming that ( X, is Gorenstein, let ω be a Gorenstein form. Then there is meromorphic g so that gω f = ω on (cid:101) X \ E . By lemma 9.11, g is the restriction of a Laurent series g ∈ O Y, [ x M ] satisfying eq. (9.4).For any v ∈ V , denote by g v the principal part of g with respect to the weight (cid:96) v . We make the Claims: (a) For any n ∈ N , g n is a monomial, that is, there is a p n ∈ M so that g n = a n x p n for some a n ∈ C ∗ .(b) If v is a vertex on a bamboo connecting n ∈ N and some other node in N ∗ , then g v = a n x p n .By (b), the exponent p = p n does not depend on n , finishing the proof since hence x p ω f is aGorenstein form.(a) is proved as follows. Set q ∈ supp( g n ) arbitrarily. We then have wt n ( g ) = (cid:96) n ( q ) , and also wt u ( g ) ≤ (cid:96) u ( q ) , for any other u , since supp( g n ) ⊂ supp( g ) . In particular, − b n wt n ( g ) + (cid:88) u ∈V n wt u ( g ) ≤ − b n (cid:96) n ( q ) + (cid:88) u ∈V n (cid:96) u ( q ) . The right hand side is sero since (cid:96) n + (cid:80) u ∈V n (cid:96) u = 0 for n ∈ N . On the other hand, by the lemma 9.11,we have wt v ( g ) = div v ( g ) for all v , thus, the left hand side above equals (div( g ) , E n ) . Furthermore,since ( g ) = ( ω ) − ( ω f ) , g does not have any zeroes or poles outside the exceptional divisor, in aneighbourhood around E n , hence (div( g ) , E n ) = (( g ) , E n ) = 0 . Therefore, the inequality above isan equality, and we have wt u ( g ) = (cid:96) u ( q ) for u ∈ V n .This fact is true for any choice of q , therefore, (cid:96) u ( q (cid:48) ) = wt u ( g ) = (cid:96) u ( q ) for any u ∈ V n and forany other choice q (cid:48) . But the vectors { (cid:96) u } u ∈V n form a generator set, hence necessarily q = q (cid:48) .For (b), assume that n and n (cid:48) ∈ N ∗ are joined by a bamboo, consisting of vertices v , . . . , v s , with v ∈ V n and v s ∈ V n (cid:48) , and v i , v i +1 neighbours for i = 1 , . . . , s − . For convenience, we set v = n and v s +1 = n (cid:48) . We start by showing that wt i ( g ) = (cid:96) i ( p n ) using induction (we replace the subscript v i by just i for legibility). Indeed, for i = 0 this is clear, and we showed in the proof of (a) that thisholds for i = 1 . For the induction step we use the recursive formulas (cid:96) i +1 − b i (cid:96) i + (cid:96) i − = 0 , wt i +1 ( g ) − b i wt i ( g ) + wt i − ( g ) = 0 . The first one holds by lemma 6.9, and the second one follows from wt i ( g ) = div i ( g ) similarly asabove, although for the case i = s , we may have to use a component of the noncompact curve E n (cid:48) .We now see that for any ≤ i ≤ s , the support of g i consists of points q ∈ M for which (cid:96) i ( q ) = (cid:96) i ( p n ) and (cid:96) i ± ( q ) ≥ (cid:96) i ± ( p n ) . But these equations are equivalent to (cid:96) n ( q ) = (cid:96) n ( p n ) and (cid:96) n (cid:48) ( q ) = (cid:96) n (cid:48) ( p n ) . Therefore, supp( g i ) = supp( g n ) for these i . (cid:3) The geometric genus and the diagonal computation sequence
In this section we construct the diagonal computation sequence, and show that it computes thegeometric genus of any Newton nondegenerate, Q -Gorenstein pointed, normal surface singularityhaving a rational homology sphere link. Any computation sequence provides an upper bound for the OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 23 geometric genus. The smallest such bound is a topological invariant, and we show that this is realizedby this diagonal sequence. This is done by showing that the diagonal computation sequence countsthe lattice points “under the diagram”, whose number is precisely the geometric genus, according tocorollary 7.4.10.1 . Discussions regarding general normal surface singularities.
Throughout this section,when not mentioned specifically, π : ( (cid:101) X, E ) → ( X, denotes a resolution of a normal surfacesingularity ( X, with exceptional divisor E , whose irreducible decomposition is E = ∪ v ∈V E v . We assume that ( X, has a rational homology sphere link ; thus E v ∼ = CP for all v ∈ V .We use the notations L , L (cid:48) and E ∗ v as in section 8. For Z = (cid:80) v r v E v with r v ∈ Q we write (cid:98) Z (cid:99) = (cid:80) v (cid:98) r v (cid:99) E v . Z K denotes the canonical cycle. Note that Z K = 0 if and only if ( X, is an ADE germ. Otherwise, it is known that in the minimal resolution, or, even in the minimal goodresolution, all the coefficients of Z K are strictly positive. However, usually this is not the case innon-minimal resolutions, i.e. in our G it is not automatically guaranteed.10.2. Lemma.
In any resolution (cid:101) X → X of a normal surface singularity with (cid:98) Z K (cid:99) ≥ we have (10.1) p g = dim C H ( (cid:101) X, O (cid:101) X ( K (cid:101) X + (cid:98) Z K (cid:99) )) H ( (cid:101) X, O (cid:101) X ( K (cid:101) X )) . Proof.
By the generalized version of Grauert–Riemenschneider vanishing we have the two vanishings(10.2) H ( (cid:101) X, O (cid:101) X ( K (cid:101) X )) = 0 , H ( (cid:101) X, O (cid:101) X ( −(cid:98) Z K (cid:99) )) = 0 . Hence, if (cid:98) Z K (cid:99) = 0 then p g = 0 too. Otherwise, from the long exact sequence of cohomology groupsassociated with → O (cid:101) X ( K (cid:101) X ) → O (cid:101) X ( K (cid:101) X + (cid:98) Z K (cid:99) ) → O (cid:98) Z K (cid:99) ( K (cid:101) X + (cid:98) Z K (cid:99) ) → , we obtain that the right hand side of eq. (10.1) equals dim H ( (cid:98) Z K (cid:99) , O (cid:98) Z K (cid:99) ( K (cid:101) X + (cid:98) Z K (cid:99) )) . By Serreduality, this equals H ( (cid:98) Z K (cid:99) , O (cid:98) Z K (cid:99) ) . Now, the short exact sequence → O (cid:101) X ( −(cid:98) Z K (cid:99) ) → O (cid:101) X → O (cid:98) Z K (cid:99) → , with the above vanishing eq. (10.2) gives H ( (cid:98) Z K (cid:99) , O (cid:98) Z K (cid:99) ) ∼ = H ( (cid:101) X, O (cid:101) X ) ∼ = C p g . (cid:3) Definition. A computation sequence is a sequence of cycles ( Z i ) ki =0 from Z K + L , Z K − (cid:98) Z K (cid:99) = Z < . . . < Z k such that(i) for all ≤ i < k there is a v ( i ) ∈ V so that Z i +1 = Z i + E v ( i ) , and(ii) Z k ≥ Z K and Z k − Z K is the union of some reduced and non-intersecting rational ( − -curvesGiven such a sequence ( Z i ) ki =0 , we define L i = O (cid:101) X ( K (cid:101) X + Z K − Z i ) , Q i = L i / L i +1 . Then Q i is a line bundle on E v ( i ) . Denote by d i its degree. Since K (cid:101) X + Z K is numerically equivalentto zero, we have d i = ( − Z i , E v ( i ) ) . In particular, since E v ( i ) ∼ = CP , we get Q i = O E v ( i ) ( − d i ) and h ( E v ( i ) , Q i ) = max { , ( − Z i , E v ( i ) ) + 1 } . . Given a computation sequence ( Z i ) i , the inclusion O (cid:101) X ( K (cid:101) X + Z K − Z k ) (cid:44) → O (cid:101) X ( K (cid:101) X ) inducesan isomorphism H ( (cid:101) X, O (cid:101) X ( K (cid:101) X + Z K − Z k ) ∼ = −→ H ( (cid:101) X, O (cid:101) X ( K (cid:101) X )) . Indeed, let
U ⊂ V be such that Z k − Z K = (cid:80) u ∈U E u . Then we have a short exact sequence → O (cid:101) X ( K (cid:101) X − E U ) → O (cid:101) X ( K (cid:101) X ) → (cid:77) O E u ( K (cid:101) X ) → , which induces an exact sequence → H ( (cid:101) X, O (cid:101) X ( K (cid:101) X + Z K − Z k ) → H ( (cid:101) X, O (cid:101) X ( K (cid:101) X )) → (cid:77) H ( E u , O E u ( K (cid:101) X )) , and the right hand side vanishes, since ( E u , K (cid:101) X ) = − − g u + b u = − . Corollary.
Let ( Z i ) ki =0 be a computation sequence. Then (10.3) p g = k − (cid:88) i =0 dim H ( (cid:101) X, L i ) H ( (cid:101) X, L i +1 ) ≤ k − (cid:88) i =0 max { , d i + 1 } . with equality if and only if the map H ( (cid:101) X, L i ) → H ( E v ( i ) , Q i ) is surjective for all ≤ i < k . (cid:3) Remark. (i) We note in particular that if there exists a computation sequence ( Z i ) ki =0 so that ( Z i , E v ( i ) ) > for all i , then p g = 0 , that is, ( X, is rational. In general, if ( Z i , E v ( i ) ) > for some i , then the inequality between the i th terms in the sums eq. (10.3) is an equality.(ii) Let S ( Z i ) be the sum (cid:80) i max { , d i + 1 } from the right hand side of eq. (10.3) associated with ( Z i ) . Then we have(10.4) p g ≤ min ( Z i ) S ( Z i ) , where the minimum is taken over all computation sequences. Note that min ( Z i ) S ( Z i ) is an invariantassociated with the topological type (graph), hence in this way we get a topological upper bound forthe geometric genus of all possible analytic types supported on a fixed topological type.On the other hand we emphasize the following facts. In general it is hard to identify a sequencewhich minimizes { S ( Z i ) } . Also, for an arbitrary fixed topological type, it is not even true thatthere exists an analytic type supported on the fixed topological type for which eq. (10.4) holds.Furthermore, it is even harder to identify those analytic structures which maximize p g , e.g., ifeq. (10.4) holds for some analytic structure, then which are these maximizing analytic structures,see e.g. [24].In the sequel our aim is the following: in our toric Newton nondegenerate case we constructcombinatorially a sequence (it will be called ‘diagonal sequence’), which satisfies eq. (10.3) withequality (in particular it minimizes { S ( Z i ) } as well). This also shows that if a topological type isrealized by a Newton nondegenerate Weil divisor, then this germ maximizes the geometric genus ofanalytic types supported by that topological type.10.7 . We recall the construction of the
Laufer operator and generalized Laufer sequences with respectto
N ⊂ V . We claim that for any cycle Z ∈ L (cid:48) , there is a smallest cycle x ( Z ) ∈ Z + L satisfying(10.5) (cid:26) ∀ n ∈ N : m n ( x ( Z )) = m n ( Z ) , ∀ v ∈ V \ N : ( x ( Z ) , E v ) ≤ . The existence and uniqueness of such an element is explained in [21] in the case when |N | = 1 and ingeneral in [16, 25, 31]. The name comes from a construction of Laufer in [17, Proposition 4.1]. Notethat x ( Z ) only depends on the multiplicities m n ( Z ) of Z for n ∈ N and the class [ Z ] ∈ H = L (cid:48) /L .The following properties hold for the operator x , assuming Z − Z ∈ L : Monotonicity: If Z ≤ Z then x ( Z ) ≤ x ( Z ) . Idempotency:
We have x ( x ( Z )) = x ( Z ) for any Z ∈ L (cid:48) . Lower bound by intersection numbers: If Z ∈ L (cid:48) and Z (cid:48) ∈ L Q so that m n ( Z ) = m n ( Z (cid:48) ) for n ∈ N and ( Z (cid:48) , E v ) ≥ for all v ∈ V \ N , then x ( Z ) ≥ Z (cid:48) . Generalized Laufer sequence:
Assume that Z ≤ x ( Z ) . First note that if ( Z, E v ) > for some v ∈ V \ N , then we have Z + E v ≤ x ( Z ) as well, similarly as in the proof of Proposition 4.1 [17]. Weclaim that there exists a generalized Laufer sequence which connects Z with x ( Z ) . It is determinedrecursively as follows. Start by setting Z = Z . Assume that we have constructed Z i . By induction,we then have Z i ≤ x ( Z ) . If ( Z, E v ) ≤ for all v ∈ V \ N then by the minimality of x ( Z ) we get Z i = x ( Z ) ; hence the construction is finished and we stop. Otherwise, there is a v ∈ V \ N so that ( Z, E v ) > . We then define Z i +1 = Z i + E v (for some choice of such v ).10.8. Remark.
The computation sequence ( Z i ) ki =0 (as in corollary 10.5), what we will construct,will have several intermediate parts formed by generalized Laufer sequences as above. Note that if Z i and Z i +1 = Z i + E v are two consecutive elements in a Laufer sequence, then − d i = ( Z i , E v ) > ,hence max { , d i + 1 } = 0 , and the comment from remark 10.6 applies: this step does not contributein the sum on the right hand side of eq. (10.3). Informally, we say that parts given by Laufersequences “do not contribute to the geometric genus”. OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 25 . The Newton nondegenerate case.
Let us consider again the resolution (cid:101) X → X of Newtonnondegenerate Weil divisor as in section 6. Let K (cid:101) X denote a canonical divisor as in section 8. Inthis section we will assume that in the dual resolution graph G we have m n ( Z K ) ≥ for any node n . This assumption will be justified in section 11.From the assumption m n ( Z K ) ≥ , valid for any node n , an immediate application of the con-struction of G from section 6 gives that Z K ≥ . Thus (cid:98) Z K (cid:99) > .10.10. Lemma. (i) x ( Z K − (cid:98) Z K (cid:99) ) ≥ Z K − (cid:98) Z K (cid:99) .(ii) Let U ⊂ V be the set of ( − -vertices appearing on bamboos joining n, n (cid:48) ∈ N ∗ with α ( (cid:96) n , (cid:96) n (cid:48) ) =1 in definition 6.4. Then x ( Z K ) = Z K + (cid:80) u ∈U E u . In particular, the sequence constructed indefinition 10.11 satisfies (ii) in definition 10.3.Proof. (i) Since x ( Z ) − Z ∈ L for any Z ∈ L (cid:48) , it is enough to show that x ( Z K − (cid:98) Z K (cid:99) ) ≥ . We cananalyse each component of G \ N independently, let G B be such a bamboo formed from E , . . . , E s ,with dual vectors in G B denoted by E ∗ i . If a ≥ and b ≥ are the multiplicities of Z K − (cid:98) Z K (cid:99) along the neighboring nodes of G B in G (with convention that a = 0 if there is only one such node),we search for a cycle x with ( x, E i ) ≤ ( aE ∗ + bE ∗ s , E i ) for all i . Thus, x − ( aE ∗ + bE ∗ s ) is in theLipman cone of G B , hence x ≥ aE ∗ + bE ∗ s ≥ .(ii) Using the lower bound by intersection numbers, we find that x ( Z K ) ≥ Z K − E + (cid:80) n ∈N E n .Since x ( Z K ) = x ( Z K − E + (cid:80) n ∈N E n ) , there exists a Laufer sequence from Z K − E + (cid:80) n ∈N E n to x ( Z K ) . Now, one verifies that the construction/algorithm of this sequence chooses each vertex v ∈ V \ ( N ∪ U ) once, and each vertex in U twice. (cid:3) Definition. A (coarse) diagonal computation sequence ( ¯ Z i ) ¯ ki =0 with respect to N is definedas follows. Start with Z = Z K − (cid:98) Z K (cid:99) , and define ¯ Z = x ( Z K − (cid:98) Z K (cid:99) ) . Assuming ¯ Z i ( i ≥ ) hasbeen defined, and that ¯ Z i | N < Z K | N , choose a ¯ v ( i ) ∈ N minimizing the ratio(10.6) n (cid:55)→ r ( n ) := m n ( ¯ Z i ) m n ( Z K − E ) , n ∈ N . Then set ¯ Z i +1 = x ( ¯ Z i + E ¯ v ( i ) ) . If ¯ Z i | N = ( Z K − E ) | N , then we record ¯ k (cid:48) = i . If ¯ Z i | N = Z K | N , thenwe stop, and set ¯ k = i .We refine the above choice as follows. Choose some node n ∈ N and define a partial order ≤ on the set N : for n, n (cid:48) ∈ N , define n ≤ n (cid:48) if n lies on the geodesic joining n (cid:48) and n (here we makeuse of the assumption that the link is a rational homology sphere, in particular, G is a tree). Whenchoosing ¯ v ( i ) , if given a choice of several nodes minimizing { r ( n ) } n , and min n { r ( n ) } < , then, wechoose ¯ v ( i ) minimal of those with respect to ( N , ≤ ) . If min n { r ( n ) } = 1 , let N (cid:48) ⊂ N be the set ofnodes n for which r ( n ) = 1 . If N (cid:48) has one element we have to chose that one. Otherwise, let G (cid:48) bethe minimal connected subgraph of G containing N (cid:48) , and we choose ¯ v ( i ) as a leaf of G (cid:48) .Note that by lemma 10.10(i), Z = Z K − (cid:98) Z K (cid:99) ≤ x ( Z K − (cid:98) Z K (cid:99) ) = ¯ Z , hence there exists a Laufersequence connecting Z with ¯ Z . Furthermore, using idempotency and monotonicity of the Lauferoperator 10.7, we find ¯ Z i + E ¯ v ( i ) = x ( ¯ Z i ) + E ¯ v ( i ) ≤ x ( ¯ Z i + E ¯ v ( i ) ) = ¯ Z i +1 . As a result, we can join ¯ Z i + E ¯ v ( i ) and ¯ Z i +1 by a Laufer sequence. This way, we obtain a computationsequence ( Z i ) i , connecting Z K − (cid:98) Z K (cid:99) with x ( Z K ) . Finally, by lemma 10.10(ii), x ( Z k ) satisfies therequirement definition 10.3(ii) too, hence corollary 10.5 applies.10.12 . For a diagonal computation sequence as above at each step, except for the step from ¯ Z i to ¯ Z i + E ¯ v ( i ) , we have d i < , we find, using lemmas 10.2 and 10.10(10.7) p g ≤ ¯ k − (cid:88) i =0 max { , ( − ¯ Z i , E ¯ v ( i ) ) + 1 } . Theorem.
Let ( X, be a normal Newton nondegenerate Weil divisor given by a function f ,with a rational homology sphere link, and assume that the polyhedron Γ + ( f ) is Q -Gorenstein pointedat p ∈ M Q . Then, a diagonal computation sequence ( Z i ) i constructed above computes the geometricgenus, that is, equality holds in eq. (10.7). In this sequel we prove the theorem under the assumption 10.9 regarding the multiplicities of Z K ,by the results of the next section this assumption can be removed.10.14. Definition.
Let n ∈ N , corresponding to the face F n ⊂ Γ( f ) . Denote by C n the convex hullof the union of F n and { p } . Set also C − n = C n \ (cid:91) n (cid:48) ≥ n C n (cid:48) , where we use the partial ordering ≤ on N defined in definition 10.11. For i = 0 , . . . , ¯ k − , let H i be the hyperplane in M R defined as the set of points q ∈ M R satisfying (cid:96) n ( q − p ) = m ¯ v ( i ) ( ¯ Z i ) . For i = 0 , . . . , ¯ k − , we set F i = C ¯ v ( i ) ∩ H i , F − i = C − ¯ v ( i ) ∩ H i . Remark.
The affine plane H i contains an affine lattice M ∩ H , that is there is an affineisomorphism H → R , inducing a bijection H ∩ M → Z . The polyhedron F i is then the image ofa lattice polyhedron with no integral integer points under a homothety with ratio in [0 , if i < ¯ k (cid:48) .These properties allow us to apply lemma 10.21 in the proof of theorem 10.13. Furthermore, thepolygon F i is always nonempty, even if F − i may be empty.10.16 . The sets C − n form a partitioning of the union of segments starting at p and ending in pointson Γ( f ) , that is, ∪ n ∈N C n . This follows from the construction as follows. The partially ordered set ( N , ≤ ) is an lower semilattice , i.e. any subset has a largest lower bound. If q ∈ ∪ n ∈N C n , and I ⊂ N is the set of nodes n for which q ∈ C n , then q ∈ C − n q , and p / ∈ C − n for n (cid:54) = n q , where n q is the largerstlower bound of I .The integral points q in the union of the sets C − n \ F n are presicely the integral points satisfying (cid:96) σ ( q ) > m σ for all σ ∈ (cid:101) (cid:52) ∗ (1 , f and (cid:96) σ ( q ) ≤ m σ for some σ ∈ (cid:52) f \ (cid:52) ∗ f . Indeed, by the rationalhomology sphere assumption, any integral point on the Newton diagram Γ( f ) must lie on the bound-ary ∂ Γ( f ) , see remark 6.5. These are the points “under the Newton diagram”; by theorem 7.3, thenumber of these points is p g . It follows from construction that the family ( F − i ∩ M ) ¯ k − i =0 forms apartition of these points. We conclude:(10.8) p g = ¯ k (cid:48) − (cid:88) i =0 | F − i ∩ M | . Definition.
For r, x ∈ R , denote by (cid:100) r (cid:101) x the smallest real number larger or equal to r andcongruent to x modulo Z . That is, (cid:100) r (cid:101) x = min { a ∈ R | a ≥ r, a ≡ x (mod Z ) } Remark.
The number (cid:100) r (cid:101) x depends on x only up to an integer. For all i , we have ¯ Z i ≡ Z K (mod L ) . In particular, given an n ∈ N , we have m n ( ¯ Z i ) ≡ m n ( Z K − E ) (mod Z ) .10.19. Lemma.
Let Z ∈ L (cid:48) and take n, n (cid:48) ∈ N ∗ connected by a bamboo, and u ∈ V a neighbour of n on this bamboo. Then (10.9) m u ( x ( Z )) = (cid:24) βm n ( Z ) + m n (cid:48) ( Z ) α (cid:25) m u ( Z ) where α = α ( (cid:96) n , (cid:96) n (cid:48) ) and β = β ( (cid:96) n , (cid:96) n (cid:48) ) (see definition 4.2 and remark 4.8). Furthermore, if for all v ∈ V lying on the bamboo joining n, n (cid:48) , we have ( Z, E v ) = 0 , then x ( Z ) = Z along the bamboo and (10.10) m u ( x ( Z )) = βm n ( Z ) + m n (cid:48) ( Z ) α . Proof.
We prove eq. (10.9), eq. (10.10) follows similarly. Let u = u , . . . , u s be the vertices on thebamboo with Euler numbers − b , . . . , − b s as in fig. 3. Set (cid:101) m = m = m n ( Z ) and (cid:101) m s +1 = m s +1 = m n (cid:48) ( Z ) . There exists a unique set of numbers (cid:101) m , . . . , (cid:101) m s ∈ Q so that the equations(10.11) (cid:101) m i − − b i (cid:101) m i + (cid:101) m i +1 = 0 , i = 1 , . . . , s OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 27 are satisfied. This follows from the fact that the intersection matrix of the bamboo is invertible over Q . In fact, it follows from [12, Lemma 20.2] that in fact, (cid:101) m = βm + m s +1 α . This, and the lower bound by intersection numbers from 10.7, implies that m u ( x ( Z )) ≥ (cid:101) m , andtherefore m u ( x ( Z )) ≥ (cid:100) (cid:101) m (cid:101) m u ( Z ) , since x ( Z ) − Z ∈ L .For the inverse inequality, we must show that there exist numbers m , . . . , m s satisfying(10.12) m i − − b i m i + m i +1 ≤ , m i ≡ m i ( Z ) (mod Z ) , for i = 1 , . . . , s , and so that m is the right hand side of eq. (10.9). Let (cid:96) n = (cid:96) , (cid:96) , . . . , (cid:96) s , (cid:96) s +1 = (cid:96) n (cid:48) be the canoncial primitive sequence as in definition 6.4, and note that β = α ( (cid:96) , (cid:96) s +1 ) . Set recursively m i = (cid:24) α ( (cid:96) i , (cid:96) s +1 ) m i − + m s +1 α ( (cid:96) i − , (cid:96) s +1 ) (cid:25) m ui ( Z ) i = 1 , . . . , s. Note that, by definition, m i ≡ m i ( Z ) . The assumption Z ∈ L (cid:48) therefore implies that the left handside of eq. (10.12) is an integer. It is then enough to prove eq. (10.12) for i = 1 . This equation isclear if s = 1 , so we assume that s > . Setting γ = α ( (cid:96) , (cid:96) s ) , we find m − (cid:101) m = (cid:24) γm + m s +1 β (cid:25) m u ( Z ) − γ (cid:101) m + (cid:101) m s +1 β = γβ ( m − (cid:101) m ) + r where ≤ r < . In order to prove eq. (10.12), we start by subtracting zero, i.e. the left hand sideof eq. (10.11). The left hand side of eq. (10.12) equals m − (cid:101) m − b ( m − (cid:101) m ) + m − (cid:101) m = (cid:18) − b + γβ (cid:19) ( m − (cid:101) m ) + r < , since γ/β < . Since the left hand side is an integer, eq. (10.12) follows. (cid:3) Lemma. If ¯ k (cid:48) ≤ i < ¯ k , then ( ¯ Z i , E ¯ v ( i ) ) > . As a result, the corresponding terms in eq. (10.7)vanish.Proof. Let u ∈ V n be a neighbour of ¯ v ( i ) . Assume first that u lies on a bamboo connecting ¯ v ( i ) and n ∈ N . We then have m ¯ v ( i ) ( ¯ Z i ) = m ¯ v ( i ) ( Z K − E ) . Furthermore, m n ( ¯ Z i ) = m n ( Z K − E ) + ε , where ε equals or . By the previous lemma, we find m u ( ¯ Z i ) = (cid:24) βm ¯ v ( i ) ( ¯ Z i ) + m n ( Z K − E ) + εα (cid:25) m u ( Z K ) = m u ( Z K − E ) + ε. with α, β as in the lemma.Next, assume that u lies on a bamboo connecting ¯ v ( i ) and n (cid:48) ∈ N ∗ \ N . Name the vertices onthe bamboo u , . . . , u s as in the proof of the previous lemma. We then have ( Z K − E, E u j ) = 0 for j = 1 , . . . , s − , and ( Z K − E, E u s ) = 1 . By the lower bound on intersection numbers, we find x ( Z K − E ) ≥ Z K − E . A Laufer sequence which computes x ( Z K − E ) from Z K − E may start with E u s , E u s − , . . . , E u . This shows that m u ( x ( Z K − E )) ≥ m u ( Z K − E ) + 1 in this case.As a result, for every u ∈ V ¯ v ( i ) , we have m u ( x ( Z K − E )) ≥ m u ( Z K − E ) , with an equality for atmost one neighbour. As a result, since ( Z K − E, E v ) = 2 − δ v we find ( ¯ Z i , E ¯ v ( i ) ) ≥ ( Z K − E, E ¯ v ( i ) ) + ( δ ¯ v ( i ) −
1) = 1 . The final statement of the lemma is now clear. (cid:3)
Lemma.
Let F ⊂ R be an integral polygon with no internal integral points. Let S , . . . , S r be the faces of F and let c j be the integral lenght of S j . Let ≤ ρ < , J ⊂ { , . . . , r } . Then let a i : R → R be the unique integral affine function whose minimal set on ρF is ρS j and this minimalvalue is λ j ∈ ] − , if j / ∈ J and λ j ∈ [ − , if j ∈ J . Set F − ρ = ρF \ ∪ j ∈ J ρS j . Then there existsan a ∈ Z satifying s (cid:88) j =1 c j a j ≡ a, | F − ρ ∩ Z | = max { , a + 1 } . Proof.
This is [31, Theroem 4.2.2]. (cid:3)
Proof of theorem 10.13.
Recall the order ≤ on the set N defined in definition 10.11. We extend thisorder in the obvious way to all of V . Also, by assumption, f is Q -pointed at the point p ∈ M Q . Fixan ≤ i ≤ ¯ k (cid:48) and set H = H i . For u ∈ V ¯ v ( i ) , define λ u = inf { (cid:96) u ( q ) | q ∈ F i } (recall that F i is nonempty, see remark 10.15) and ν u = (cid:40) λ u + 1 if u ≤ ¯ v ( i ) and λ u ∈ Z , (cid:100) λ u (cid:101) else . Define the affine functions a u : H → R , a u = (cid:96) u | H − ν u . By construction, these are primitive integralfunctions on H with respect to the affine lattice H ∩ M . It now follows from lemma 10.21 that thereis an a ∈ Z so that (cid:80) u a u ≡ a and | F − i ∩ M | = max { , a + 1 } .On the other hand, we claim that ν u − (cid:96) u ( p ) ≤ m u ( ¯ Z i ) for u ∈ V ¯ v ( i ) . Using lemma 6.9, and thedefinition of H i , i.e. (cid:96) ¯ v ( i ) ( q − p ) | H = m ¯ v ( i ) ( ¯ Z i ) for q ∈ H , it follows that a = (cid:88) u a u ( q ) = (cid:88) u (cid:96) u ( q − p ) − ( ν u − (cid:96) u ( p )) ≥ b ¯ v ( i ) (cid:96) ¯ v ( i ) ( q − p ) − (cid:88) u m u ( ¯ Z i ) = ( − ¯ Z i , E ¯ v ( i ) ) . where q is any element of H . As a result, using eq. (10.7) and lemma 10.20, as well as eq. (10.8),we have p g = ¯ k (cid:48) − (cid:88) i =0 | F − i ∩ M | ≥ ¯ k − (cid:88) i =0 max { , ( − ¯ Z i , E ¯ v ( i ) ) + 1 } ≥ p g , and so these inequalities are in fact equalities.We are left with proving the claim ν u ≤ m u ( ¯ Z i ) + (cid:96) u ( p ) for u ∈ V ¯ v ( i ) . Fix u , and let n ∈ N ∗ sothat u lies on a bamboo connecting ¯ v ( i ) and n . Let S = F ¯ v ( i ) ∩ F n . Then S is the minimal set of (cid:96) u on F ¯ v ( i ) , i.e., S = F u . Let A be the affine hull of S ∪ { p } . Since the two affine functions (cid:96) ¯ v ( i ) − (cid:96) ¯ v ( i ) ( p ) m ¯ v ( i ) ( Z K − E ) , (cid:96) n − (cid:96) n ( p ) m n ( Z K − E ) , both take value on p and on S , by theorem 9.6(iii), they conincide on A . Let r = m ¯ v ( i ) ( ¯ Z i ) m ¯ v ( i ) ( Z K − E ) Using the minimality of eq. (10.6), we get for any q ∈ p + r ( S − p ) ⊂ H ∩ A (10.13) (cid:96) n ( q − p ) m n ( Z K − E ) = (cid:96) ¯ v ( i ) ( q − p ) m ¯ v ( i ) ( Z K − E ) = m ¯ v ( i ) ( ¯ Z i ) m ¯ v ( i ) ( Z K − E ) ≤ m n ( ¯ Z i ) m n ( Z K − E ) , and so (cid:96) n ( q − p ) ≤ m n ( ¯ Z i ) . In the case when n ≤ ¯ v ( i ) , or equivalently, u ≤ ¯ v ( i ) , this inequality isstrict. It follows, using lemma 10.19, that m u ( ¯ Z i ) = (cid:24) β ( (cid:96) ¯ v ( i ) , (cid:96) n ) m ¯ v ( i ) ( ¯ Z i ) + m n ( ¯ Z i ) α ( (cid:96) ¯ v ( i ) , (cid:96) n ) (cid:25) m u ( ¯ Z i ) ≥ β ( (cid:96) ¯ v ( i ) , (cid:96) n ) (cid:96) ¯ v ( i ) ( q − p ) + (cid:96) n ( q − p ) α ( (cid:96) ¯ v ( i ) , (cid:96) n )= (cid:96) u ( q − p )= λ u − (cid:96) u ( p ) . (10.14)Therefore, since m u ( ¯ Z i ) ≡ m u ( Z K ) ≡ − (cid:96) u ( p ) (mod Z ) , we find m u ( ¯ Z i ) ≥ (cid:100) λ u (cid:101) − (cid:96) u ( p ) . OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 29
This proves the claim, unless u ≤ ¯ v ( i ) and λ u ∈ Z . In that case, the numbers (cid:96) ¯ v ( i ) ( q ) and (cid:96) u ( q ) = λ u are both integers. Since (cid:96) n , (cid:96) u form a part of an integral basis of N = M ∨ , we can assume that q ∈ M , hence, β ( (cid:96) ¯ v ( i ) , (cid:96) n ) (cid:96) ¯ v ( i ) ( q − p ) + (cid:96) n ( q − p ) α ( (cid:96) ¯ v ( i ) , (cid:96) n ) = (cid:96) u ( q − p ) ≡ − (cid:96) u ( p ) ≡ m u ( Z K ) ≡ m u ( ¯ Z i ) (mod Z ) . As a result, since we have a strict inequality m n ( ¯ Z i ) > (cid:96) n ( q − p ) we get a strict inequality ineq. (10.14) as well. Therefore, we have m u ( ¯ Z i ) > λ u − (cid:96) u ( p ) and m u ( ¯ Z i ) ≡ λ u − (cid:96) u ( p ) (mod Z ) , and so m u ( ¯ Z i ) ≥ λ u − (cid:96) u ( p ) + 1 = ν u − (cid:96) u ( p ) , which finishes the proof of the claim. (cid:3) Removing B -facets In this section we consider only surface singularities, i.e. we assume that r = 3 . We consider removable B -facets of two dimensional Newton diagrams and show that they can be removed with-out affecting certain invariants of nondegenerate Weil divisors. This is stated in proposition 11.7.In parallel we also prove proposition 11.13, which allows us to assume that the divisor Z K − E onthe resolution provided by Oka’s algorithm has nonnegative multiplicities on nodes, cf. 10.9 and thesentence after theorem 10.13. Similar computations are given in [7], providing a stronger result inthe case of a hypersurface singularity in C with rational homology sphere link.The concept of a B -facet appears in [11] in the case of hypersurfaces in K r , where K is a p -adicfield, and is further studied in [18, 6].11.1. Definition.
Let F ⊂ Γ( f ) be a compact facet, i.e. of dimension . Then F is a B -facet if F has exactly vertices p , p , p so that there is a σ ∈ (cid:52) (1)Σ so that m σ = (cid:96) σ ( p ) = (cid:96) σ ( p ) = (cid:96) σ ( p ) − .A B -facet F is removable if furthermore, the segment [ p , p ] is contained in the boundary ∂ Γ( f ) of Γ( f ) . p p p σF σ F σ + Figure 6.
On the left we have a Newton diagram in R ≥ with a removable B facet F . To the left, we see the -skeleton of the dual fan, and an intersection witha hyperplane. In this example we have (cid:96) σ ( p ) = (cid:96) σ ( p ) = 0 and (cid:96) σ ( p ) = 1 .11.2. Definition.
Let T ( f ) be closure in N R of the union of cones in (cid:52) f which correspond tocompact facets of Γ + ( f ) which have dimension > . This is the tropicalization of f . We say that Σ is generated by the tropicalization of f , if Σ is generated as a cone by the set T ( f ) .Let Σ (cid:48) be the cone generated by T ( f ) . This is a finitely generated rational strictly convex cone,and if ( X, is not rational, then Σ (cid:48) has dimension r = 3 . This cone induces an affine toric variety Y (cid:48) = Y Σ (cid:48) , and the function f defines a Weil divisor ( X (cid:48) , ⊂ ( Y (cid:48) , . Furthermore, the inclusion Σ (cid:48) ⊂ Σ induces a morphism Y (cid:48) → Y , which restricts to a morphism ( X (cid:48) , → ( X, .11.3. Remark.
The closure of T ( f ) in a certain partial compactification of N R is called the localtropicalization of ( X, [30]. x xy z T ( f ) M R N R Figure 7.
Here,
Σ = R ≥ is the positive octant, and f ( x, y, z ) = x + xy + z isthe E singularity in normal form. In this case, T ( f ) does not generate Σ , but thecone generated by (2 , , , (0 , , and (0 , , .11.4. Lemma.
Let σ ∈ (cid:101) (cid:52) f . Then the orbit O σ intersects (cid:101) X if and only if σ ⊂ T ( f ) .Proof. The orbit O σ is an affine variety O σ = Spec( C [ M ( σ )]) (recall M ( σ ) = M ∩ σ ⊥ ) , and if p σ isan element of the affine hull of F σ , then x − p σ f σ ∈ C [ M ( σ )] and (cid:101) X ∩ O σ ∼ = Spec (cid:18) C [ M ( σ )]( x − p σ f σ ) (cid:19) . Therefore, (cid:101) X ∩ O σ is empty if and only if x − p σ f σ is a unit in C [ M ( σ )] , which is equivalent to f σ being a monomial, i.e. dim F σ = 0 . (cid:3) Lemma.
Let ( X, and ( X (cid:48) , be as in definition 11.2. If ( X, is normal, then the morphism ( X (cid:48) , → ( X, is an isomorphism.Proof. We can assume that the smooth subdivision (cid:101) (cid:52) f subdivides the cone Σ (cid:48) , so that we get asubdivision (cid:101) (cid:52) (cid:48) f = (cid:101) (cid:52) f | Σ (cid:48) of the cone Σ (cid:48) . Let (cid:101) Y (cid:48) be the corresponding toric variety. Let (cid:52) T ( f ) be thefan consisting of cones σ ∈ (cid:101) (cid:52) f which are contained in T ( f ) . We then get open inclusions Y T ( f ) ⊂ (cid:101) Y (cid:48) ⊂ (cid:101) Y where Y T ( f ) is the toric variety associated with the fan (cid:52) T ( f ) .It follows from lemma 11.4 that the strict transforms (cid:101) X and (cid:101) X (cid:48) of X and X (cid:48) , respectively, arecontained in Y T ( f ) , and so (cid:101) X (cid:48) = (cid:101) X . As a result, X (cid:48) \ { } ∼ = (cid:101) X (cid:48) \ π − (0) = (cid:101) X \ π − (0) ∼ = X \ { } .Since ( X, is normal, the morphism ( X (cid:48) , → ( X, is an isomorphism. (cid:3) . Assume that F ⊂ Γ( f ) is a removable B -face, and let σ ∈ (cid:52) (1)Σ and p i be as in definition 11.1.If F is the only facet of Γ( f ) , then we leave as an exercise to show that the graph G is equivalentto a string of rational curves, and so ( X, is rational. We will always assume that F is not theonly facet of Γ( f ) . There exists an element of Σ ◦ which is constant on the segment [ p , p ] (e.g. thenormal vector to F ). As a result, the boundary ∂ Σ intersects the hyperplane of elements (cid:96) ∈ N R which are constant on [ p , p ] in two rays, σ + and σ − , where (cid:96) ∈ σ + satisfies (cid:96) | [ p ,p ] ≡ max F (cid:96) , and (cid:96) ∈ σ − satisfies (cid:96) | [ p ,p ] ≡ min F (cid:96) .Let (cid:96) + ∈ N be a primitive generator of σ + , set m + = max F (cid:96) and define ¯ f ( x ) = (cid:88) { a p x p | p ∈ M, (cid:96) + ( p ) ≥ m + } , where a p are the coefficients of f as in eq. (3.1). Let ( ¯ X, be the Weil divisor defined by ¯ f .We get a Newton polyhedron Γ + ( ¯ f ) , from which we calculate invariants of ( ¯ X, as described inprevious sections. It follows from this construction that Γ( ¯ f ) = Γ( f ) \ F , and that ¯ f is Newtonnondegenerate.Now, assume that Σ is generated by the tropicalization of f . Let σ and σ ∈ (cid:52) (1) f be therays corresponding to the noncompact faces of Γ + ( f ) containing the segments [ p , p ] and [ p , p ] ,respectively. Let (cid:96) , (cid:96) be primitive generators of σ , σ . By construction, and the above assumptionthat Σ is generated by T ( f ) , we have R ≥ (cid:104) (cid:96) , (cid:96) (cid:105) ⊂ ∂ Σ , and so (cid:96) + ∈ R ≥ (cid:104) (cid:96) , (cid:96) (cid:105) ∈ (cid:52) f . OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 31
In fact, we have (cid:96) + = (cid:96) + t(cid:96) where t = (cid:96) ( p − p ) . Indeed, (cid:96) + is the unique positive linearcombination of (cid:96) and (cid:96) which vanishes on p − p , and is primitive. Since (cid:96) = (cid:96) σ , by definition of F , and since (cid:96) ( p ) = (cid:96) ( p ) , we have ( (cid:96) + t(cid:96) )( p − p ) = (cid:96) ( p − p ) + (cid:96) ( p − p ) · (cid:96) ( p − p ) = (cid:96) ( p − p ) − (cid:96) ( p − p ) = 0 . Furthermore, we have (cid:96) ( p − p ) = 0 and (cid:96) ( p − p ) = 1 , and so by remark 4.3, (cid:96) , (cid:96) form a partof an integral basis, which implies that (cid:96) + t(cid:96) is primitive.Now, define t (cid:48) as the combinatorial length of the segment [ p , p ] . We have t (cid:48) | t and via Oka’salgorithm (definition 6.4), this segment corresponds to t (cid:48) bamboos in G , each consisting of a single ( − -curve, whereas [ p , p ] corresponds to one bamboo with determinant t/t (cid:48) .11.7. Proposition.
Let f , F and ¯ f be as above, and assume that f is Newton nondegenerate.Assume also that Σ is generated by the tropicalization T ( f ) as described in definition 11.2. Then(i) ¯ f is Newton nondegenerate.(ii) Γ( ¯ f ) = Γ( f ) \ F .(iii) The singularities ( X, and ( ¯ X, have diffeomorphic links.(iv) The singularities ( X, and ( ¯ X, have equal geometric genera and δ -invariants.(v) If ( X, is normal, then ( ¯ X, is normal.(vi) If f is Q -Gorenstein-pointed at p ∈ M Q , then so is ¯ f . In particular, if ( X, is Gorenstein,then ( ¯ X, is also Gorenstein.Proof. (i) and (ii) follow from definition.We now prove (iii). We have G , the output of Oka’s algorithm for the Newton polyhedron Γ + ( f ) ,and ¯ G , the output of Oka’s algorithm for Γ + ( ¯ f ) . Let σ F ∈ (cid:52) f be the ray dual to F and let F (cid:48) bethe unique face of Γ + ( f ) adjacent to F , i.e. F (cid:48) ∩ F = [ p , p ] . Then σ F ⊂ R ≥ (cid:104) (cid:96) F (cid:48) , (cid:96) + (cid:105) ∈ (cid:52) ¯ f , andwe can subdivide the canonical subdivision of R ≥ (cid:104) (cid:96) (cid:48) F , (cid:96) + (cid:105) so that we can assume that σ F ∈ (cid:101) (cid:52) ¯ f .We can therefore identify vertices v F of G and ¯ G corresponding to the same ray σ F ∈ (cid:101) (cid:52) (1) f and σ F ∈ (cid:101) (cid:52) (1)¯ f . It is then clear from construction that the components of G \ v F and ¯ G \ v F in thedirection of v F (cid:48) are isomorphic. After blowing down the ( − -curves corresponding to the segment [ p , p ] , we must show (cid:96) The two bamboos joining (cid:96) F with (cid:96) + on one hand, and with (cid:96) on the other, are isomorphic. (cid:96) The vertex v F has the same Euler number in G and in ¯ G . − v v v F v F (cid:48) v F (cid:48) v + G ∗ ¯ G ∗ ¯ v F u uu (cid:48) ¯ u (cid:48) Figure 8.
The ( − -curve to the left is blown down, so that the two graphs G and ¯ G , obtained by deleting v , v , v + and their adjacent edges, look topologically thesame. To the right, the bamboo connecting ¯ v F (cid:48) and v + corresponds to a subdivisionof the cone generated by (cid:96) + and (cid:96) F (cid:48) which contains the ray generated by (cid:96) F . For the first of these, we prove that α ( (cid:96) F , (cid:96) + ) = α ( (cid:96) F , (cid:96) ) , β ( (cid:96) F , (cid:96) + ) = β ( (cid:96) F , (cid:96) ) . We calculate α ( (cid:96) F , (cid:96) + ) as the greatest common divisor of maximal minors of the matrix havingcoordinate vectors for (cid:96) F and (cid:96) + = (cid:96) + t(cid:96) as rows. But α ( (cid:96) F , (cid:96) ) = t/t (cid:48) , and so adding a multipleof t to (cid:96) does not modify the greatest common divisor of these determinants, hence α ( (cid:96) F , (cid:96) + ) = α ( (cid:96) F , (cid:96) + t(cid:96) ) = α ( (cid:96) F , (cid:96) ) .The invariant β ( (cid:96) F , (cid:96) + ) can be calculated as the unique number ≤ β < α ( (cid:96) F , (cid:96) + ) so that β(cid:96) F + (cid:96) + is a multiple of α ( (cid:96) F , (cid:96) + ) . On the other hand, we find, setting β = β ( (cid:96) F , (cid:96) ) and α = α ( (cid:96) F , (cid:96) + ) = α ( (cid:96) F , (cid:96) ) = t/t (cid:48) , β(cid:96) F + (cid:96) + α = β(cid:96) F + (cid:96) + t(cid:96) α = β(cid:96) F + (cid:96) α + t (cid:48) (cid:96) ∈ N. Finally, we show that v F has the same Euler number in the graphs G and ¯ G . Denote these by − b F and − ¯ b F . After blowing down the ( − curves associated with the segment [ p , p ] , the vertex v F has two neighbors in either graph G or ¯ G . Denote by v − and ¯ v − the neighbor of v F containedin the same component of G \ v F and ¯ G \ v F as v F (cid:48) . It is then clear that (cid:96) v − = (cid:96) ¯ v − .Denote by u, ¯ v the neighbours of v F , ¯ v F in the direction of v , v + , respectively, and u (cid:48) , ¯ u (cid:48) the otherneighbours, as in fig. 8. Then we have (cid:96) u (cid:48) = (cid:96) ¯ u (cid:48) and (cid:96) u = β(cid:96) F + (cid:96) α , (cid:96) ¯ u = β(cid:96) F + (cid:96) + α = (cid:96) u + t (cid:48) (cid:96) , where α, β are as above. The two numbers − b F and − ¯ b F are identified by lemma 6.9 − b F (cid:96) F + (cid:96) u + (cid:96) u (cid:48) + t (cid:48) (cid:96) = 0 , − ¯ b F (cid:96) F + (cid:96) ¯ u + (cid:96) ¯ u (cid:48) = 0 , which leads to their equality.Next, we prove (iv) and (v). By theorem 7.3, it suffices to show that Γ + ( f ) \ (Σ ∨ + q ) , Γ + ( ¯ f ) \ (Σ ∨ + q ) have the same cohomology for all q ∈ M . By shifting Γ + ( f ) , we simplify the following proof byassuming q = 0 . The inclusion Γ( f ) \ Σ ∨ ⊂ Γ + ( f ) \ Σ ∨ is a homotopy equivalence. Indeed, one can construct a suitable vectorfied on Γ + ( f ) \ Σ ∨ pointingin the direction of − Σ ∨ , whose trajectories end up in Γ( f ) \ Σ , thus giving a homotopy inverse tothe above inclusion.Now, let K be the union of faces of Γ( f ) which do not intersect Σ ∨ . By lemma 7.7, the inclusion K ⊂ Γ( f ) \ Σ is a homotopy equivalence. Define ¯ K similarly, using ¯ f . Thus it suffices to prove that (cid:101) H i ( K, ¯ K ; Z ) vanish for all i . By excision, this is equivalent to showing(11.1) ∀ i ∈ Z ≥ : (cid:101) H i ( K ∩ F, ¯ K ∩ F ; Z ) = 0 . If Σ ∨ does not intersect the face F , then K ∩ F = F = ¯ K ∩ F . Also, if p ∈ Σ ∨ , then K ∩ F = ¯ K ∩ F .In either case, eq. (11.1) holds. We can therefore assume that p ∈ K and F (cid:54)⊂ K . With theseassumptions at hand, it is then enough to prove that excactly one of the segements [ p , p ] and [ p , p ] is contained in K , i.e. it cannot happen that either both or neither is contained in K .Let A be the affine hull of F n , i.e. the hyperplane in M R defined by (cid:96) n = m n , and let C = Σ ∨ ∩ A .Define a point r ∈ A by (cid:96) ( r ) = 0 , (cid:96) ( r ) = 0 , (cid:96) n ( r ) = m n . This is well defined, since the functions (cid:96) , (cid:96) , (cid:96) n are linearly independent. Then C is a convexpolygon in A , and r is a vertex of C . Furthermore, r is the unique point in C where both functions (cid:96) | C and (cid:96) | C take their minimal values.If neither of the segments [ p , p ] , [ p , p ] are contained in K , i.e. both intersect Σ ∨ , then we canchoose r ∈ C ∩ [ p , p ] and r ∈ C ∩ [ p , p ] . Furthermore, we have (cid:96) ( r ) ≤ (cid:96) ( r ) = (cid:96) ( p ) , and (cid:96) ( r ) ≤ (cid:96) ( r ) = (cid:96) ( p ) . Therefore, p is in the convex hull of r, r , r , and so p ∈ C , contrary tothe assumption p ∈ K .Next, assume that both segments [ p , p ] , [ p , p ] are contained in K . We start by showing thatin this case, we have r ∈ F n . By assumption, we can choose r (cid:48) ∈ C ∩ F n . OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 33 F n rr (cid:48) (cid:96) < m (cid:96) (cid:96) > m (cid:96) p p p (cid:96) > m (cid:96) (cid:96) < m (cid:96) A Figure 9.
The segment [ r (cid:48) , r ] intersects neither [ p , p ] nor [ p , p ] .We have (cid:96) ( r (cid:48) ) > m (cid:96) . One verifies (see fig. 9) that if (cid:96) ( r ) ≤ m (cid:96) , then we would have (cid:96) ( r (cid:48) ) <(cid:96) ( r ) , but r is a minimum for (cid:96) | C . Therefore, we can assume that (cid:96) ( r ) > m (cid:96) , similarly, (cid:96) ( r ) >m (cid:96) . It follows, since C ∩ F (cid:54) = ∅ , that r ∈ F , so we can assume that r (cid:48) = r . But, since r / ∈ [ p , p ] ∪ [ p , p ] , we find (cid:96) ( p ) < (cid:96) ( r ) < (cid:96) ( p ) = (cid:96) ( p ) + 1 , and so (cid:96) ( r ) / ∈ Z . But this is a contradiction, since (cid:96) ( r ) = (cid:96) ( q ) ∈ Z .Next we prove (vi). Assume that Γ + ( f ) is Q -Gorenstein pointed at p ∈ M Q . It suffices to showthat (cid:96) + ( p ) = ¯ m (cid:96) + + 1 , where ¯ m (cid:96) + is the minimal value of (cid:96) + on Γ + ( ¯ f ) . We immediately find ¯ m (cid:96) + = (cid:96) + ( p ) = (cid:96) ( p ) + t(cid:96) ( p ) = m (cid:96) + t ( m (cid:96) + 1) = (cid:96) ( p ) − t(cid:96) ( p ) = (cid:96) + ( p ) − . (cid:3) Example.
Consider the cone
Σ = R ≥ and the function f ( x, y, z ) = x + xy + z + y z, which defines a nonrational singularity ( X, . In this case, Γ( f ) has a B -facet F = conv { (1 , , , (0 , , , (0 , , } , corresponding to a node n ∈ N . The normal vector to F is (19 , , and eq. (8.2) gives m n ( Z K − E ) = − . By the above computations, removing the monomial y z from f gives another singularity withthe same link and geometric genus, but Z K − E is nonnegative on the other node. After removing F we find ¯ f ( x, y, z ) = x + xy + z . Note that Σ is generated by the tropicalization of f , but the tropicalization of ¯ f generates the cone R ≥ (cid:104) (5 , , , (0 , , , (0 , , (cid:105) . x xy (1 , , , , , , , ,
0) (15 , , z (5 , , F (19 , , y z y z Figure 10.
A diagram with a B -facet F and its dual. The dotted line to the rightreplaces its two neighbouring segments if the B -facet is removed. . In what follows, we connect the above construction with the coefficients of Z K − E . Weintroduce a simplified graph, whose vertices are the nodes of G . whose vertices are the nodes of G ,and a bamboo of G connecting two nodes of G is replaced in G N by an edge. Then G N is a tree,with an edge connecting n, n (cid:48) if and only if F n and F (cid:48) n intersect in a segment (of length ). Recallthat a leaf of a tree is a vertex with exactly one neighbour. If we assume that |N | > , then we seethat the following are equivalent, since G N is a tree: (cid:96) n ∈ N is a leaf in G N , (cid:96) Γ( f ) \ F n is connected, (cid:96) all edges of F n , except for one, lie on the boundary ∂ Γ( f ) of the Newton diagram.If |N | = 1 , then there is a unique n ∈ N , and Γ( f ) = F , in particular, ∂ Γ( f ) = ∂F n . Finally, if |N | = 0 , and if we assume that ( X, is normal, then ( X, is rational. length = t length = t length = s Figure 11.
A big triangle, a small triangle of type t = 3 , and a trapezoid of type ( t, s ) = (4 , .The following lemma is elementary:11.10. Lemma.
Let F be an integral polyhedron in R , having no integral interior points. Then, upto an integral affine automorphism of R , F is one one the following: (cid:96) Big triangle
The convex hull of (0 , , (2 , , (0 , . (cid:96) Small triangle of type t The convex hull of (0 , , ( t, , (0 , . (cid:96) Trapezoid of type ( t, s ) The convex hull of (0 , , ( t, , (0 , , ( s, , where t, s ∈ Z , t ≥ s > and t > . (cid:3) Lemma.
Assume that ( X, is normal, Gorenstein-pointed at p ∈ M , and not rational. If n ∈ N is a leaf in G N and m n ( Z K − E ) < , then F n is a removable B -facet of Γ( f ) (See 11.9 forthe definition of G N ).Proof. By assumption, F n has two adjacent edges contained in ∂ Γ( f ) , say [ q , q ] and [ q , q ] . Let F , F be the noncompact faces of Γ + ( f ) containing the segments [ q , q ] and [ q , q ] , respectively,and let (cid:96) , (cid:96) ∈ ∂ Σ be the primitive functions having F , F as minimal sets on Γ + ( f ) , denote theseminimal values by m (cid:96) , m (cid:96) .Let l = length([ q , q ]) and α = (cid:96) ( q − q ) /l and l = length([ q , q ]) and α = (cid:96) ( q − q ) /l .Then, the bamboos corresponding to the segements [ q , q ] and [ q , q ] have determininats α , α ,see remark 4.3.Assume first that F n is a small triangle of type t , that the segment [ q , q ] has length t , and that α = 1 . This implies that F n is a removable B -facet.Otherwise, let A be the affine hull of F n . If F n is a big triangle, a trapezoid, or a small triangleas above, but with α > , then the square { q ∈ A | m (cid:96) ≤ (cid:96) ( q ) ≤ m (cid:96) + 1 , m (cid:96) ≤ (cid:96) ( q ) ≤ m (cid:96) + 1 } is contained in F n . In particular, its vertex q , the unique point in A satisfying (cid:96) i ( q ) = m (cid:96) i + 1 for i = 1 , , is contained in F n . The set R = { q ∈ Σ ∨ | (cid:96) i ( q ) = 0 , i = 1 , } OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 35 is a one dimensional face of Σ ∨ (here we use the condition that Σ is generated by the tropicalizationof ( X, ). By our assumption m n ≤ (cid:96) n ( p ) we have p ∈ q + R ◦ ⊂ Γ + ( f ) ◦ , contradicting theassumption that ( X, is not rational. (cid:3) Proposition.
Assume that ( X, is normal, Gorenstein-pointed at p ∈ M , and not rational.If there is an n ∈ N so that m n ( Z K − E ) < , then Γ( f ) has a removable B -facet.Proof. If n is a leaf in G N (see 11.9), then F n is removable by lemma 11.11. So let us assume that n is not a leaf in G N , i.e. that Γ( f ) \ F n is disconnected. The inclusion Γ( f ) ◦ \ F n ⊂ Γ ∗ + ( f ) ◦ \ ( { (cid:96) n ≤ m n } ∪ Γ + ( f ) ◦ ) is a strong homotopy retract (here we set Γ( f ) ◦ = Γ( f ) \ ∂ Γ( f ) ). In particular, the right hand side isdisconnected as well. But it follows from our assumptions that the point p is in the right hand sideabove. Let C be a component of Γ( f ) \ F n contained in a component of the right hand side whichdoes not contain p . Then, for any n (cid:48) so that F n (cid:48) ⊂ C we have (cid:96) n (cid:48) ( p ) > m n (cid:48) , i.e. m n (cid:48) ( Z K − E ) < .Let G C be the induced subgraph of G N having vertices n (cid:48) for F n (cid:48) ⊂ C . This graph is a nonemptytree, and so has either exactly one vertex, or at has least two leaves. In the first case, the uniquevertex n (cid:48) of G C is a leaf of G . In the second case, G C has at least two leaves, so we can choosea leaf n (cid:48) of G C which is not adjacent to n in G . In either case, F n (cid:48) is a removable B -facet bylemma 11.11. (cid:3) Proposition.
Assume that f defines a normal Newton nondegenerate Weil divisor ( X, ,which is not rational. Then there exists a normal Newton nondegenerate Weil divisor ( ¯ X, , definedby a function ¯ f and a cone Σ (cid:48) (possibly different than Σ ) satisfying the following conditions: (cid:96) ( ¯ X, and ( X, have diffeomorphic links. (cid:96) p g ( ¯ X,
0) = p g ( X, . (cid:96) If ( X, is Gorenstein or pointed at p ∈ M Q , then so is ( ¯ X, . (cid:96) If F n ⊂ Γ + ( ¯ f ) is a compact facet, then m n ( Z K − E ) ≥ .In fact, Γ( ¯ f ) is the union of those facets F n of Γ( f ) for which m n ( Z K − E ) ≥ .Proof. By lemma 11.5, we can assume that Σ is generated by T ( f ) , since ( X, is normal (seedefinition 11.2). The result therefore follows, using induction on the number of facets of Γ( f ) , andpropositions 11.7 and 11.12 below. (cid:3) Examples
Example.
Let N = M = Z and let a, b, c ∈ N be natural numbers with no common factor,and let ≤ r < s ∈ N be coprime with s ≤ rc . Take Σ ∨ = R ≥ (cid:42) ( ra, , − s )( 0 , rb, − s )( 0 , , (cid:43) , f = x a + x b + x c . The cone Σ is then generated by (cid:96) = (1 , , , (cid:96) = (0 , , , (cid:96) = 1gcd( ab, s ) ( bs, as, abr ) . Corresponding to these, we have irreducible invariant divisors D , D , D ⊂ Y and multiplicities m = 0 , m = 0 , m = abs gcd( ab, s ) . The Newton diagram Γ( f ) consists of a single face with normal vector (cid:96) = ( bc, ac, ab ) and m = abc .Fulton shows in 3.4 of [13] that the group of Weil divisors modulo linear equivalence on Y is generatedby D , D , D , and that (cid:80) j =1 a i D i is Cartier if and only if there is a p = ( p , p , p ) ∈ M = Z sothat a j = (cid:96) j ( p ) for j = 1 , , .In our case, X is equivalent to − (cid:80) i =1 m i D i = − m D . Therefore, if X is Cartier, then there isa p = ( p , p , p ) ∈ M so that (cid:96) i ( p ) = m i . Therefore, we find p = p = 0 , and abr gcd( ab, s ) p = abs gcd( ab, s ) . Therefore, X is Cartier if and only if r | s , i.e. r = 1 . M R N R (5,3,10)(1,0,0) (0,1,0)(35,21,15) y z x Figure 12.
In the above examples, we have a = 3 , b = 5 , c = 7 , r = 2 and s = 3 .The cone Σ is generated by the vectors (1 , , , (0 , , and (5 , , . Furthermore, (35 , , is the normal vector to the unique facet of Γ( f ) .12.2. Example.
In [24], Némethi and Okuma analyse upper and lower bounds for the geometricgenus of singularities with a specific topological type, namely, whose link is given by the plumbinggraph in fig. 13. − − − − − − − Figure 13.
A resolution graphThey show that for this graph, the path lattice cohomology is , but that the maximal geometricgenus among analytic structures with this topological type is . As a result, this graph is not thetopological type of a Newton nondegenerate Weil divisor in a toric affine space.On the other hand, this topological type is realized by the complete intersection given by thesplice equations X = (cid:8) z ∈ C (cid:12)(cid:12) z z + z + z = z + z + z z = 0 (cid:9) . This singularity is in fact a Newton nondegenerate isolated complete intersection [28]. As a result, themethods of section 10 do not generalize in the most straightforward way to Newton nondegeneratecomplete intersections.
References [1] Altmann, Klaus. Toric Q -Gorenstein singularities. arXiv:alg-geom/9403003.[2] Altmann, Klaus; Ploog, David. Displaying the cohomology of toric line bundles. Izv. Math. , 84(4):683–693, 2020.[3] Arnold, Vladimir I.; Gusein-Zade, Sabir M.; Varchenko, Alexander N.
Singularities of differentiable maps. Volume2.
Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2012.[4] Aroca, Fuensanta; Gómez-Morales, Mirna; Shabbir, Khurram. Torical modification of Newton non-degenerateideals.
Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM , 107(1):221–239, 2013.[5] Artin, Michael. On isolated rational singularities of surfaces.
Am. J. Math. , 88:129–136, 1966.[6] Bories, Bart; Veys, Willem.
Igusa’s p -adic local zeta function and the monodromy conjecture for non-degeneratesurface singularities. , volume 1145. Providence, RI: American Mathematical Society (AMS), 2016.[7] Braun, Gábor; Némethi, András. Invariants of Newton non-degenerate surface singularities. Compos. Math. ,143(4):1003–1036, 2007.[8] Bruns, Winfried; Gubeladze, Joseph.
Polytopes, rings, and K-theory.
New York, NY: Springer, 2009.[9] Bruns, Winfried; Herzog, Jürgen.
Cohen-Macaulay rings. Rev. ed , volume 39. Cambridge: Cambridge UniversityPress, rev. ed. edition, 1998.
OCAL NEWTON NONDEGENERATE WEIL DIVISORS IN TORIC VARIETIES 37 [10] Danilov, V. I. The geometry of toric varieties.
Uspekhi Mat. Nauk , 33(2(200)):85–134, 247, 1978.[11] Denef, Jan. Poles of p -adic complex powers and Newton polyhedra. Nieuw Arch. Wiskd., IV. Ser. , 13(3):289–295,1995.[12] Eisenbud, David; Neumann, Walter D.
Three-dimensional link theory and invariants of plane curve singularities. ,volume 110 of
Annals of Mathematics Studies . Princeton University Press, 1985.[13] Fulton, William.
Introduction to toric varieties , volume 131 of
Annals of Mathematics Studies . Princeton Uni-versity Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry.[14] Karras, Ulrich. Local cohomology along exceptional sets.
Math. Ann. , 275:673–682, 1986.[15] Kouchnirenko, A. G. Polyèdres de Newton et nombres de Milnor.
Invent. Math. , 32:1–31, 1976.[16] László, Tamás.
Lattice cohomology and Seiberg–Witten invariants of normal surface singularities . Ph.D. thesis,Central European University, 2013. ArXiv:1310.3682.[17] Laufer, Henry B. On rational singularities.
Am. J. Math. , 94:597–608, 1972.[18] Lemahieu, Ann; Van Proeyen, Lise. Monodromy conjecture for nondegenerate surface singularities.
Trans. Am.Math. Soc. , 363(9):4801–4829, 2011.[19] Merle, Michel; Teissier, Bernard. Conditions d’adjonction. (D’après Du Val). In
Séminaire sur les singularitésdes surfaces , volume 777 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin Heiderberg New York, 1980.[20] Némethi, András. Resolution graphs of some surface singularities. I: Cyclic coverings. In
Singularities in algebraicand analytic geometry. Papers from the AMS special session, San Antonio, TX, USA, January 13-14, 1999 ,89–128. Providence, RI: American Mathematical Society (AMS), 2000.[21] Némethi, András. On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds.
Geom. Topol. ,9:991–1042, 2005.[22] Némethi, András. Pairs of invariants of surface singularities. In
Proceedings of the international congress ofmathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Volume II. Invited lectures , 749–779.Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM), 2018.[23] Némethi, András; Nicolaescu, Liviu I. Seiberg–Witten invariants and surface singularities.
Geom. Topol. , 6:269–328, 2002.[24] Némethi, András; Okuma, Tomohiro. Analytic singularities supported by a specific integral homology spherelink.
Methods Appl. Anal. , 24(2):303–320, 2017.[25] Némethi, András; Sigurðsson, Baldur. The geometric genus of hypersurface singularities.
J. Eur. Math. Soc.(JEMS) , 18(4):825–851, 2016.[26] Oda, Tadao.
Convex bodies and algebraic geometry. An introduction to the theory of toric varieties.
Berlin etc.:Springer-Verlag, 1988.[27] Oka, Mutsuo. On the resolution of the hypersurface singularities. In
Complex analytic singularities , volume 8 of
Adv. Stud. Pure Math. , 405–436. North-Holland, Amsterdam, 1987.[28] Oka, Mutsuo. Principal zeta-function of non-degenerate complete intersection singularity.
J. Fac. Sci., Univ.Tokyo, Sect. I A , 37(1):11–32, 1990.[29] Popescu-Pampu, Patrick. The geometry of continued fractions and the topology of surface singularities. In
Singu-larities in geometry and topology 2004 , volume 46 of
Adv. Stud. Pure Math. , 119–195. Math. Soc. Japan, Tokyo,2007.[30] Popescu-Pampu, Patrick; Stepanov, Dmitry. Local tropicalization. In
Algebraic and combinatorial aspects of trop-ical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathe-matical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011 , 253–316. Providence, RI: AmericanMathematical Society (AMS), 2013.[31] Sigurðsson, Baldur.
The geometric genus and the Seiberg–Witten invariant of Newton nondegenerate surfacesingularities . Ph.D. thesis, Central European University, 2015. ArXiv:1601.02572.[32] Varchenko, Alexander N. Zeta-function of monodromy and Newton’s diagram.
Invent. Math. , 37:253–262, 1976.
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