aa r X i v : . [ m a t h . AG ] F e b ON INTRINSIC NEGATIVE CURVES
ANTONIO LAFACE AND LUCA UGAGLIA
Abstract.
Let K be an algebraically closed field of characteristic 0. A curve of ( K ∗ ) arising from a Laurent polynomial in two variables is intrinsic negative if its tropical compactification has negative self-intersection. The aim of thisnote is to start a systematic study of these curves and to relate them with theproblem of computing Seshadri constants of toric surfaces. Introduction
Following the work of Ananya, Gonz´alez, Karu [16], Kurano [20] and KuranoMatsuoka [21], we define a class of curves on the blowing-up of toric surfaces at ageneral point. Let f be a Laurent polynomial in two variables and let Γ ⊆ ( K ∗ ) beits zero locus. The normal fan to the Newton polygon of f defines a toric variety P such that the compactification of Γ is contained in the smooth locus of P . Sucha compactification is called tropical , see [26]. Denote by X ∶= Bl e P the blowing-upof P at the image e of ( , ) and let C be the strict transform of the compactifiedcurve. We say that C is an intrinsic negative curve (resp. non positive ) if C < C ≤ Proposition 1.
There exist infinite families of non-positive intrinsic curves, whoseNewton polygons are listed in the following tablevertices of ∆ lw ( ∆ ) C g ( C ) (i) [ m
10 1 m ] m ≥ − (ii) [ m − m m − m −
20 0 1 m m − ] m ≥ − (iii) [ m − m − m m −
10 1 m m m − m − m − ] m = k ≥ − (iv) [ m − m m − m −
20 0 1 m m − ] m ≥ Date : February 19, 2021.2010
Mathematics Subject Classification.
Primary 14M25; Secondary 14C20.
Key words and phrases.
Toric surfaces, Seshadri constants.Both authors have been partially supported by Proyecto FONDECYT Regular n. 1190777. (v) [ m − m m − m −
30 0 1 m m − ] m = k + ≥ P , anample class H and a point x ∈ P , the Seshadri constant of H at x can be defined as ε ( H, x ) ∶= inf x ∈ C H ⋅ C mult x ( C ) where the infimum is taken over all irreducible curves through x . The problem offinding Seshadri constants of algebraic surfaces have been widely studied (see forinstance [6,14,15,24] and the references therein). When P is a toric surface there arethree possibilities for x ∈ P : either the point is torus-invariant, or it lies on a torus-invariant curve, or it is general. In the first two cases, since the blowing-up Bl x P admits the action of a torus of dimension two and one respectively, it is possible todescribe the effective cone (see [11,25] and [4, § §
4] and [19, § ε ( H ∆ , e ) , where ( P ∆ , H ∆ ) is the toric pair defined by ∆ (see § e ∈ P ∆ is a general point. In order to state the result, given a non-negativeinteger m denote by L ∆ ( m ) the linear system of Laurent polynomials whose expo-nents are integer points of ∆ and such that all the partial derivatives up to order m − ( , ) . If we denote by vol ( ∆ ) the normalized volume of ∆ (that istwice its euclidean area), and by lw ( ∆ ) its lattice width (see for instance [23, Def.1.8]), we have the following (the first inequality is well known [1, Thm. 0.1] and[19], but we state it anyway for the sake of completeness). Proposition 2.
Let ∆ ⊆ Q be a lattice polygon, let ( P ∆ , H ∆ ) be the correspondingtoric pair and let ε ∶ = ε ( H ∆ , e ) be the Seshadri constant at e ∈ P . Then the followinghold. (i) ε ≤ lw ( ∆ ) . (ii) If vol ( ∆ ) > lw ( ∆ ) then ε ∈ Q . (iii) If there exists an m ∈ N such that vol ( ∆ ) ≤ m and L ∆ ( m ) ≠ ∅ , then ε ∈ Q . (iv) If moreover m ≤ lw ( ∆ ) , then ε ≤ vol ( ∆ )/ m , and the equality holds if L ∆ ( m ) contains an irreducible curve. We remark that Proposition 2 provides in some cases (like e.g. [17, Example 5.7])an alternative proof for the rationality of the Seshadri constant of a toric surface ata general point. Moreover it allows to compute the exact value of the Seshadri con-stant ε ( H ∆ , e ) when ( X ∆ , H ∆ ) is the toric pair associated to the Newton polygonof an intrinsic non-positive curve. Corollary 1.
Let f ∈ K [ u ± , v ± ] be a Laurent polynomial with Newton polygon ∆ and multiplicity m at ( , ) , such that the corresponding intrinsic curve C ⊆ X ∆ isnon-positive, i.e. C ≤ . Then the Seshadri constant of the ample divisor H ∆ of N INTRINSIC NEGATIVE CURVES 3 the toric surface P ∆ at a general point e ∈ P ∆ is ε = vol ( ∆ ) m . In particular the polygons of the infinite families appearing in Proposition 1 haveSeshadri constant ε = vol ( ∆ )/ lw ( ∆ ) ∈ Q . The paper is structured as follows. In §
1, after recalling some definitions andresults about toric varieties and lattice polytopes we introduce intrinsic curves andwe prove some preliminaries result. In § § Intrinsic curves
Let us first recall some definitions and set some notations we are going to usethroughout this note.Given a lattice polytope ∆ ⊆ Q n , i.e. a polytope whose vertices have integercoordinates, we can define a pair ( P ∆ , H ∆ ) consisting of a toric variety P ∆ togetherwith a very ample divisor H ∆ . The toric variety is the normalization of the closureof the image of the following monomial morphism g ∆ ∶ ( K ∗ ) n → P ∣ ∆ ∩ Z n ∣− , u ↦ [ u w ∶ w ∈ ∆ ∩ Z n ] , where u = ( u , . . . , u n ) ∈ ( K ∗ ) n . It is possible to show that the action of thetorus ( K ∗ ) n on itself extends to an action on P ∆ and that the subset of primetorus-invariant divisors is finite and in bijection with the set of facets of ∆. Let D , . . . , D r be such divisors and let v , . . . , v r be the inward normal vectors to thefacets of ∆. Each v i defines a linear form Q n → Q by w ↦ w ⋅ v i and the very ampledivisor is H ∆ ∶ = − r ∑ i = min w ∈ ∆ { w ⋅ v i } D i . In what follows we will often use the notation g, P and H , omitting the subscriptwhen it is clear from the context. We recall that if ∆ is a very ample polytope [11,Def. 2.2.17] then the closure of the image of g ∆ is a normal variety by [11, Thm.2.3.1] and thus it coincides with P ∆ . Moreover, by [11, Cor. 2.2.19], ∆ is veryample if n = n =
2, i.e. ∆ ⊆ Q is a lattice polygon, sothat P is a normal toric surface. We will denote by e ∈ P the image via g of theneutral element of the torus, by π ∶ X → P the blowing up of P at e and by E theexceptional divisor. Given an m ∈ N we will denote by L ∆ ( m ) the sublinear systemof ∣ H ∣ consisting of sections having multiplicity at least m at e . Definition 1.1.
Let f ∈ K [ u ± , v ± ] be an irreducible Laurent polynomial, let∆ ⊆ Q be the Newton polygon of f , i.e. the convex hull of its exponents, and letΓ ⊆ P ∆ be the closure of V ( f ) ⊆ ( K ∗ ) . We say that the strict transform C ⊆ X ∆ of Γ is the intrinsic curve defined by f and that C is: ● an intrinsic negative (resp. non-positive ) curve if C < C ≤ ● an intrinsic ( − n ) -curve if C = − n < p a ( C ) = A. LAFACE AND L. UGAGLIA ● expected in X ∆ ′ , with ∆ ⊆ ∆ ′ if ∣ ∆ ′ ∩ Z ∣ > ( m + ) .We remark that, with the notation above, Γ ⊆ P is an element of the very amplelinear series ∣ H ∣ and Γ ∈ L ∆ ( m ) if f has multiplicity at least m at ( , ) , that isall the partial derivatives of f up to order m − ( , ) . Moreover if themultiplicity is m then the strict transform C ⊆ X of Γ is a Cartier divisor such that(1.1) C = vol ( ∆ ) − m p a ( C ) = ( vol ( ∆ ) − ∣ ∂ ∆ ∩ Z ∣ + m − m ) + , see for instance [9, § Newton polygon of C .Our first result is about the characterization of Newton polygons of expectednon-positive curves. Proposition 1.2.
Let C be an intrinsic expected non-positive curve with Newtonpolygon ∆ and multiplicity m at e . Then one of the following holds: ● vol ( ∆ ) = m and ∣ ∂ ∆ ∩ Z ∣ = m ; ● vol ( ∆ ) = m and ∣ ∂ ∆ ∩ Z ∣ = m + ; ● vol ( ∆ ) = m − and ∣ ∂ ∆ ∩ Z ∣ = m + .In particular C ∈ { − , } and p a ( C ) ∈ { , } .Proof. Let us denote by b ∶ = ∣ ∂ ∆ ∩ Z ∣ the number of boundary lattice points of∆ and by i ∶ = ∣ ∆ ∩ Z ∣ − b the number of interior lattice points. Recall that byPick’s formula vol ( ∆ ) = i + b −
2. Being C expected and non-positive we have ∣ ∆ ∩ Z ∣ ≥ ( m + ) + ( ∆ ) ≤ m . By (1.1), the non-negativity of the arithmeticgenus of C gives ( vol ( ∆ ) − b + m − m ) + ≥
0. The three inequalities in terms of i and b are 2 i + b ≥ m ( m + ) , 2 i + b − ≤ m , 2 i − m + m ≥
0. From these onededuces that one of the following holds: { b = mi = m − m + { b = m + i = m − m . { b = m + i = m − m Since C = vol ( ∆ ) − m and C ⋅ K = m − b , in the first two cases we have C = p a ( C ) = C = − p a ( C ) = (cid:3) Proposition 1.3.
All the non-equivalent polygons for intrinsic non-positive curvesof multiplicity ≤ are the following. m ∆1234 Proof.
We use the database of polygons with small volume [5] to analyze all thepolygons with volume at most 15 and such that vol ( ∆ ) − m ≤
0. For each such poly-gon ∆ we compute L ∆ ( m ) , where m ≤
4, with the aid of the function
FindCurves of the Magma library:
N INTRINSIC NEGATIVE CURVES 5 https://github.com/alaface/non-polyhedral/blob/master/lib.m
We take only the pairs ( ∆ , m ) such that L ∆ ( m ) contains exactly one element,which is irreducible. Finally we check that in each of these cases the Newtonpolygon coincides with ∆. (cid:3) Remark 1.4.
The above intrinsic curves are all expected. In all but the last casethey are intrinsic ( − ) -curves, in the last case the curve has self-intersection 0 andgenus 1. The smallest value of m for an unexpected intrinsic negative curve is 5.The lattice polygon ∆ is the followingOne has ∣ ∂ ∆ ∩ Z ∣ = m − ∣ ∂ ∆ ∩ Z ∣ = ( m + ) which imply that the correspondingcurve has arithmetic genus 1. The curve is defined by the Laurent polynomial − uv + uv + u v − u v + u v + u v − u v − u v + u v + u v − u v − u v , which is the unique one whose Newton polygon is contained in ∆ and has multiplic-ity 5 at ( , ) . Its strict transform in X ∆ is smooth of genus 1 and self-intersection − ( ∆ , m ) . Definition 1.5.
Let ∆ be a lattice polygon, m a positive integer, and set p a ∶ = ( vol ( ∆ ) − ∣ ∂ ∆ ∩ Z ∣ + m − m ) +
1. We say that ( ∆ , m ) is: ● numerically negative (resp. non positive ) if vol ( ∆ ) − m < ≤ ); ● a ( − n ) - pair if vol ( ∆ ) − m = − n < p a = ● expected if ∣ ∆ ∩ Z ∣ > ( m + ) . Remark 1.6.
Clearly, if C is an intrinsic negative curve, then the pair ( ∆ , m ) ,consisting of the Newton polygon of C and the multiplicity of Γ = π ( C ) at e , isnumerically negative. On the other hand, if a pair ( ∆ , m ) is numerically negative,in general there does not exist an intrinsic negative curve associated to it. Indeed,first of all it can happen that L ∆ ( m ) is empty (see Example 1.7). Furthermore,even if ( ∆ , m ) is expected (so that L ∆ ( m ) is not empty), in some cases it containsonly reducible curves (see Example 1.8). Example 1.7.
Consider the following polygon ∆We have that vol ( ∆ ) =
14 and ∣ ∂ ∆ ∩ Z ∣ =
4, so that the pair ( ∆ , ) is numericallya ( − ) -pair. A direct computation shows that L ∆ ( ) = ∅ , so that there does notexists an intrinsic ( − ) -curve with Newton polygon ∆ and multiplicity 4. Example 1.8.
The polygon ∆, whose Minkowski decomposition ∆ = ∆ + ∆ isgiven in the below picture A. LAFACE AND L. UGAGLIA = +has vol ( ∆ ) = ∣ ∂ ∆ ∩ Z ∣ = ∣ ∆ ∩ Z ∣ = ( + ) +
1, so that ( ∆ , ) is an expected ( − ) -pair. The only element in L ∆ ( ) is defined by ( u v + uv − uv + ) ⋅ ( u v − u v − u v + u v − u v + u v − u v − u v − u v + u v − u v + u v + u v + uv − uv + ) . The factorization implies the Minkowski decomposition ∆ = ∆ + ∆ . The polygon∆ corresponds to an intrinsic ( − ) -curve C , while ∆ corresponds to an intrinsiccurve C of self-intersection 0 and genus 1. By Proposition 1.12 it follows that C ⋅ C = Remark 1.9.
Finally we observe that if the pair ( ∆ , m ) is numerically non pos-itive and L ∆ ( m ) contains an irreducible curve Γ, then a Laurent polynomial f ofΓ ∩ ( K ∗ ) defines an intrinsic non positive curve. Indeed, by definition the stricttransform C ⊆ X ∆ of Γ satisfies C = vol ( ∆ ) − m ≤
0. Moreover, since the Newtonpolygon ∆ ′ of f is contained in ∆, we also have that vol ( ∆ ′ ) − m ≤ vol ( ∆ ) − m ≤ ′ is strictly contained in ∆, then the self intersection of theintrinsic curve is strictly smaller than C (see Example 1.10). Example 1.10.
Let ∆ and ∆ ′ be the following polygons, respectively from left torightOne has vol ( ∆ ) = , ∣ ∂ ∆ ∩ Z ∣ = ∣ ∆ ∩ Z ∣ = ( + ) +
1, so that ( ∆ , ) is anexpected ( − ) -pair. The linear system L ∆ ( ) contains a unique irreducible curvedefined by the following polynomial f ∶ = − u v + u v − u v + u v − u v − u v + u v + u v − u v + u v + u v + u v − u v + u v + u v − u v + u v − u v + u v + uv − , where we used the colour red to denote the monomials corresponding to verticesof ∆. The Newton polygon of f is ∆ ′ , since u is the only monomial (correspondingto a lattice point of ∆) that does not appear in f . In particular, vol ( ∆ ′ ) =
34 and ∣ ∂ ∆ ∩ Z ∣ =
6, so that f defines an intrinsic (unexpected) ( − ) -curve. Definition 1.11.
Given two polygons ∆ , ∆ their mixed volume is:vol ( ∆ , ∆ ) ∶ = ( vol ( ∆ + ∆ ) − vol ( ∆ ) − vol ( ∆ )) . We conclude the section by showing how the mixed volume of two lattice polygonsrelates with the intersection product of curves on a toric variety whose fan refinesthe normal fans of the two polygons.
N INTRINSIC NEGATIVE CURVES 7
Proposition 1.12.
Let ( ∆ , m ) , ( ∆ , m ) , be two pairs, each of which consistsof a lattice polygon together with a positive integer. Assume that L ∆ i ( m i ) is non-empty and let C i ⊆ X ∆ i be the strict transform of a curve in the linear system. Let X be a surface which admits birational morphisms φ i ∶ X → X ∆ i for i = , . Then φ ∗ ( C ) ⋅ φ ∗ ( C ) = vol ( ∆ , ∆ ) − m m . Proof.
Let π i ∶ X ∆ i → P ∆ i be the blowing-up at e ∈ P ∆ i . First of all observe that C i is a Cartier divisor of X ∆ i because H i ∶ = π i ( C i ) is very ample on P ∆ i . It followsthat the pullback φ ∗ i is defined on C i , so that the left hand side of the formulais well-defined. The intersection product φ ∗ ( C ) ⋅ φ ∗ ( C ) does not depend on thesurface X because all such surfaces differ by exceptional divisors, which have zerointersection product with the pullbacks of C and C . We can then choose X ∶ = X ∆ to be the blowing-up of P ∆ at the general point e , where ∆ ∶ = ∆ + ∆ . Since H i is very ample on P ∆ i , its pullback is base point free in P ∆ . By Bertini’s theoremthe general elements of these two linear systems intersect transversely at distinctpoints which, without loss of generality, we can assume to be contained in ( K ∗ ) . ByBernstein-Kushnirenko theorem the number of these intersections is vol ( ∆ , ∆ ) ,so that the statement follows after taking into account the intersections of the twocurves at e ∈ P ∆ . (cid:3) Infinite families
In this section we construct infinite families of intrinsic negative curves. First ofall we produce infinite families of toric surfaces, each of which corresponds to anintrinsic negative curve. Then, in Example 2.4, we construct an infinite family ofintrinsic negative curves on a given toric surface.
Lemma 2.1.
Let f , f , f , f ∈ K [ t ] and let m be the maximal degree of the fourpolynomials. Assume that f − f = f − f , the polynomials f and f are coprimeand deg ( f − f ) = m . Then the image of the following rational map P / / ( K ∗ ) , t ↦ ( f / f , f / f ) has multiplicity m at ( , ) .Proof. Since f and f are coprime and f − f = f − f , we have that f and f are coprime too. Moreover, deg ( f − f ) = m and K algebraically closed implythat f − f has m roots. Any root of f − f is also a root of f − f , so that weconclude that there are m values of t (counting multiplicities) whose image is thepoint ( , ) . (cid:3) Proof of Proposition 1.
First of all, each polygon ∆ of type (v) satisfies lw ( ∆ ) = m ,vol ( ∆ ) = m and ∣ ∂ ∆ ∩ Z ∣ = m , so that the pair ( ∆ , m ) is numerically 0 and expected(in particular it has arithmetic genus 1). Moreover, in [9, Section 6] it has beenshown that if we set m = k +
4, for each k ≥ f , f , f , f ∶ = f − f + f given in the following table. f f f A. LAFACE AND L. UGAGLIA (i) − m ∑ i = t i t m (ii) ( m − ) t − ( m − ) −( t − ) t m − −( t − ) ( t m − + m − ∑ i = ( m − − i ) t i ) (iii) a k − ( t − ) t k − ( t − a )( t − a ) a t k − ( t − a ) a , a ∶ = k − k − (iv) 2 t − ( − t ) t m − −( t − ) ( m − ∑ i = t i − ) These polynomials satisfy the hypotheses of Lemma 2.1, so that the image of themap ϕ ( t ) = ( f / f , f / f ) has a point of multiplicity m at e . In order to concludewe have to show that in each case the Newton polygon is the one given in the firstcolumn of the table within the proposition. To this aim we will use [13, Thm. 1.1]which, given a parametric curve Γ ⊆ ( K ∗ ) , provides a description of the normal fanof the Newton polygon of Γ together with the length of the edges, in terms of thezeroes of the four polynomials f , . . . , f . For the sake of completeness we explainin detail case (i). In this case the map ϕ is defined by ϕ ( t ) = ( − t ∑ m − i = t i , − t m ∑ m − i = t i ) . Since ϕ satisfies ord ( ϕ ) = ( − , m ) , ord ∞ ( ϕ ) = ( m, − ) and ord q i ( ϕ ) = ( − , − ) , forall the m − q , . . . , q m − of ∑ m − i = t i , and these are the only values of t forwhich ord ( ϕ ) does not vanish, by [13, Thm. 1.1], the rays of the normal fan of theNewton polygon of ϕ ( P ) are ( − , m ) , ( m, − ) and ( − , − ) . Moreover, the firsttwo rays correspond to two edges of lattice length 1 while the third one has length m −
1. We conclude that the Newton polygon has vertices ( , ) , ( m, ) , ( , m ) . (cid:3) Remark 2.2.
The triangles of type (i) in Proposition 1 are indeed equivalentto the ones with vertices ( , ) , ( m − , ) , ( m, m + ) , i.e. IT ( m − , ) in thenotation of [2, Thm. 1.1.A]. Therefore, as a byproduct of Proposition 1 we obtainan alternative (short) proof of [3, Thm. 1.1].Observe that for each infinite family of Proposition 1, the slopes of (some of)the edges change with m , so that also the toric surfaces change. We are now goingto give an example of an infinite family of negative curves lying on the blowing-upat of a fixed toric surface. First of all we recall a construction from [9]. Givenan expected lattice polygon ∆ ⊆ Q of width m ∶ = lw ( ∆ ) , with vol ( ∆ ) = m and ∣ ∂ ∆ ∩ Z ∣ = m , if L ∆ ( m ) contains an unique irreducible element, then its stricttransform C ⊆ X ∶ = X ∆ is a curve of arithmetic genus one with C =
0. Whenever C is smooth we denote by res ∶ Pic ( X ) → Pic ( C ) N INTRINSIC NEGATIVE CURVES 9 the pullback induced by the inclusion. It is not difficult to show that the image ofthe above map is contained in Pic ( C )( Q ) . If res ( C ) ∈ Pic ( C ) is non-torsion then,by [9, Sec. 3], the divisor K X + C is linearly equivalent to an effective divisor whosesupport can be contracted by a birational morphism φ ∶ X → Y . The surface Y has at most Du Val singularities and nef anticanonical divisor − K Y ∼ C (here withabuse of notation we denote by the same letter C a curve which lives in differentbirational surfaces and is disjoint from the exceptional locus). Lemma 2.3. If Pic ( Y ) has rank three then K ⊥ Y ∩ Eff ( Y ) = Q ≥ ⋅ [ C ] .Proof. The class [ C ] spans an extremal ray of Eff ( Y ) because the curves contractedby φ are disjoint from C and thus res ( C ) is non-torsion also on Y . As a consequenceEff ( Y ) is non-polyhedral by [9, Lem. 3.3]. By [9, Lem. 3.14] the minimal resolutionof singularities π ∶ Z → Y is a smooth rational surface Z of Picard rank 10, nefanticanonical class − K Z and non-polyhedral effective cone Eff ( Z ) . Observe thatthe root sublattice of Pic ( Z ) spanned by classes of ( − ) -curves over singularities of Y has rank R =
7. Assume now that D is an effective divisor such that D ⋅ K Y = D is push-forward of an effective divisor D ′ of Z with D ′ ⋅ K Z =
0. Being − K Z nef, by adjunction D ′ = ∑ i a i C i + nC , where each C i is a ( − ) -curve and a i , n ≥ ( C i ) = i and the fact that Eff ( Y ) isnon-polyhedral we conclude that all the C i are contracted by π . Thus D is linearlyequivalent to a positive multiple of C . (cid:3) We are now going to consider a particular lattice polygon ∆ satisfying the aboveconditions (it is number 24 in [9, Table 3]) and we are going to show that the blowingup of the corresponding toric surface contains infinitely many negative curves (seeRemark 2.5).
Proposition 2.4.
Let X be the blowing-up at a general point of the toric surface P , defined by the following lattice polygon ∆ ⊆ Q D D D D D D Then there is a birational morphism φ ∶ X → Y onto a rational surface Y of Picardrank three with only Du Val singularities. Moreover if D , . . . , D are the pullbacksof the prime invariant divisors of P , ordered according to the picture, the pushfor-ward on Y of the divisor E k ∶ = ( k − ) D + ( k − k − ) D + ( k − k + ) D +( k − k + ) D − ( k − k ) E is linearly equivalent to a ( − ) -curve for any integer k ≠ .Proof. Let C be the curve of X defined by the unique element in L ∆ ( ) , which hasequation f ∶ = − + v + uv − u v − uv + u v + u v + uv + u v − u v + u v − u v − uv − u v + u v + u v − u v + u v + uv + u v − u v − uv , where we marked with a red label the monomials corresponding to the vertices of∆. Let us denote by v , . . . , v the primitive generators of the rays of the normalfan to ∆, which are the columns of the following matrix P ∶ = ( − − − − − − ) . By [17, Prop. 3.1] the divisor π ( C ) is linearly equivalent to ∑ i = − a i π ( D i ) , where a i is the minimum of the linear form w ↦ w ⋅ v i on ∆. A direct calculation gives C ∼ D + D + D + D − E. By (1.1) the curve C has self-intersection C = y + y = x + x , labelled 43.a1 in the LMFDB database. Its Mordell-Weil group Pic ( C )( Q ) is free of rank one so that res ( C ) is either trivial or non-torsion. The first possibilityis ruled out by the fact that dim ∣ C ∣ = dim L ∆ ( ) = C in X (see [9, Lem. 3.2] for details). Thus res ( C ) is non-torsionand so [ C ] spans an extremal ray of Eff ( X ) . By [9, Cor. 3.12] the divisor K X + C is linearly equivalent to an effective divisor whose support can be contracted. Thiscontraction is the morphism φ in the statement. We claim that K X + C ∼ C + C , where C ∼ D + D − E and C ∼ D + D − E are the strict transforms of the twoone-parameter subgroups corresponding to the width directions ( , ) and ( , ) of∆. To prove the claim it suffices to observe that the divisor K X + C − C − C ∼ − D + D + D − D − D is principal, being a linear combination of the rowsof the above matrix P . Using the intersection matrix of D , . . . , D , E ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ − / / − − / / / /
189 1 /
27 0 00 0 0 1 / − /
27 1 01 / − / − ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ we can see that C i ⋅ C = C ⋅ C = i = , E k has integer intersectionproduct with all the D i , in particular it is a Cartier divisor. Moreover, E k ⋅ C = E k ⋅ C = E k = E k ⋅ K = − , so that by Riemann-Roch E k is effective. To prove that it is irreducible we proceedas follows. Since X has Picard rank 5 and φ ∶ X → Y contracts C ∪ C , the surface Y has Picard rank 3. The push-forward φ ∗ E k is an effective and Cartier divisor,because we are contracting curves which have intersection product zero with E k .Moreover, since − K Y is nef and − K Y ⋅ φ ∗ E k = , by Lemma 2.3 we deduce that either φ ∗ E k is irreducible or φ ∗ E k = D + nC , with D irreducible and n >
0. Since D is Cartier, both D and D ⋅ K Y are integers,and moreover, being − K Y nef, by the genus formula D ≥ −
2. The case D = − D would be in K ⊥ Y . Thus D = D ⋅ K Y = − N INTRINSIC NEGATIVE CURVES 11 − = φ ∗ E k = D + nD ⋅ C = − + n > φ ∗ E k is linearly equivalent to a ( − ) -curve. (cid:3) Remark 2.5.
A priori the curve E k could be reducible of the form E k = Γ k + a C + a C , with Γ k irreducible and a , a ≥
0. So we are only showing the existence ofinfinitely many negative curves Γ k on X , which are not necessarily ( − ) -curves.Moreover even if E k = Γ k , the Newton polygon of E k does not necessarily coin-cide with the Riemann Roch polygon ∆ k of the curve E k . So that E k could bea ( − ) -curve but not an intrinsic one. The Riemann-Roch polygon has verticescorresponding to the columns of the following matrix ( k − k k − k + k − k − k − k k + k k − k k − k + k − k − k − k ) . In particular, if we set m = k − k , we have that vol ( ∆ k ) = m − ∣ ∂ ∆ k ∩ Z ∣ = m + ( ∆ k ) = m , so that ( ∆ k , m ) is numerically a ( − ) -pair. For smallvalues of k it is possible to check, with the help of the software Magma [8], thatthe Newton polygon of the ( − ) -curve is indeed ∆ k , but for general k we are notable to prove it. Therefore Γ k is not necessarily an intrinsic ( − ) -curve, but it isanyway an intrinsic negative curve (see Remark 1.6).3. Seshadri constants
In this section we first prove Proposition 2 and Corollary 1, and then we discusssome consequences on the study of the effective cone of the blowing up of weightedprojective planes.We will need the following preliminary result about Seshadri constants on pro-jective surfaces.
Lemma 3.1.
Let Y be a projective surface, H an ample divisor of Y , and let π ∶ X → Y be the blowing-up of Y at p ∈ Y with exceptional divisor E . (i) If there is a positive integer m such that π ∗ H − mE is the class of aneffective curve C = ∑ ri = a i C i with C ≤ , then ε ( H, p ) = min i { π ∗ H ⋅ C i E ⋅ C i } ≤ m + C m . (ii) If furthermore C is irreducible, then ε ( H, p ) = m + C m .Proof. We prove (i). Let ε ∶ = ε ( H, p ) . Observe that we can write C + ( m − ε ) E ⋅ C = ( π ∗ H − mE + ( m − ε ) E ) ⋅ C = ( π ∗ H − εE ) ⋅ C ≥
0, and, since E ⋅ C = m , we get ε ≤ m + C m ≤ m. If C is nef then ε ≥ m , so that ε = m and C =
0. This implies C ⋅ C i =
0, and hence ε = π ∗ H ⋅ C i / E ⋅ C i for any i , proving the statement in this case. If C is not nef then ε < m . If α is such that ε < α < m then π ∗ H − αE is effective and non-nef. Let C ′ bean irreducible curve such that ( π ∗ H − αE ) ⋅ C ′ <
0, then ( π ∗ H − mE ) ⋅ C ′ < C ′ = C i for some i . Since α can be chosen arbitrarily close to ε , we concludethat ( π ∗ H − εE ) ⋅ C j = j , and the statement follows. Statement (ii) is animmediate consequence of (i) and the equality π ∗ H ⋅ C = C + m . (cid:3) Proof of Proposition 2.
We prove (i). Let v ∈ N be a width direction, that islw v ( ∆ ) = lw ( ∆ ) and let C v ⊆ X ∆ be the strict transform of the one-parametersubgroup of the torus defined by v . If µ > lw ( ∆ ) , then ( π ∗ H − µE ) ⋅ C v <
0, so that π ∗ H − µE is not nef. This proves the statement.We prove (ii). Observe that ( π ∗ H − εE ) ≥ ( π ∗ H − lw ( ∆ ) E ) = vol ( ∆ ) − lw ( ∆ ) >
0, where the first inequality is by (i). By the Riemann-Roch theorem, the class of π ∗ H − εE is in the interior of the effective cone Eff ( X ∆ ) . It follows that the Seshadriconstant is computed by a curve C ⊆ X ∆ . From ( π ∗ H − εE ) ⋅ C = ε is a rational number (see also [22, Rem. 2.3]).Statements (iii) and (iv) are consequence of Lemma 3.1 and the fact that C = vol ( ∆ ) − m . (cid:3) Proof of Corollary 1.
Observe that by hypothesis vol ( ∆ ) − m = C ≤
0, so thatthe hypothesis (iii) of Proposition 2 is satisfied. Moreover, being C irreducible wehave lw ( ∆ ) − m = C ⋅ C v ≥
0, where C v is the strict transform of the one-parametersubgroup of the torus defined by the width direction v . Thus also hypothesis (iv)of Proposition 2 is satisfied and the statement follows. (cid:3) Remark 3.2.
Observe that the best lower bound for ε we can get from [19,Thm. 1.3] in the case of a toric surface is either the width lw ( ∆ ) , or the biggestlength of a segment inside ∆. For instance, if ∆ is a triangle of type (i) in Propo-sition 1, consider the projection π ∶ Q → Q onto the second factor. If we take thefiber ∆ ∩ π − ( ) , by [19, Thm. 1.3] we have the inequality ε ( H, e ) ≥ min { m, m − / m } = m − / m. Since vol ( ∆ ) = m −
1, by Proposition 2 (iii), we also have the inequality ε ( H, e ) ≤ ( m − )/ m , so that we can conclude that ε ( H, e ) is indeed equal to m − / m , noneed of showing that there exists an irreducible curve C ∈ L ∆ ( m ) . mm mπ We also remark that in the remaining cases of Proposition 1 the bound given by [19,Thm. 1.3] is not sharp.In the same vein, if the lattice polygon ∆ contains a segment of lattice lengthlw ( ∆ ) , then [19, Thm. 1.3] gives the bound ε ( H, e ) ≥ lw ( ∆ ) . But since by Propo-sition 2 (i) we also have the opposite inequality, we can immediately conclude thatthe Seshadri constant ε ( H, e ) is indeed equal to lw ( ∆ ) .For instance, this shows that for any m ≥
4, the polygon with vertices ( , ) , ( , ) , ( m, ) , ( , m ) corresponds to a toric surface with Seshadri constant ε ( H, e ) = m (even if it is not hard to find a parametrisation as we did with the families ofProposition 1). N INTRINSIC NEGATIVE CURVES 13
Weighted projective planes.
We briefly recall an open problem about theexistence of certain irreducible curves in weighted projective planes, and its rela-tion with intrinsic curves. Let a, b, c be three positive pairwise coprime integers,let P ( a, b, c ) be the corresponding weighted projective plane and let π ∶ X ( a, b, c ) → P ( a, b, c ) be the the blowing-up at the general point e ∶ = ( , , ) with exceptionaldivisor E . The divisor class group of X ∶ = X ( a, b, c ) is free of rank 2 and the effec-tive cone Eff ( X ) is in general unknown. By the Riemann-Roch theorem, Eff ( X ) contains the positive light cone Q (shaded region) with extremal rays generated by R ± = π ∗ H ± √ abc E . Eπ ∗ HR − R + The question is whether Eff ( X ) is bounded by the R -divisor R − , so that ε ( H, e ) = /√ abc , or by the class of a negative curve (lying below the ray R − ). In manyexamples (see for instance [16] and [18]) the existence of the negative curve hasbeen proved, but in general the question is still open, and in fact it is conjecturedthat for some triples a, b, c (such as 9 , ,
13) that negative curve does not exist. Weremark that proving this conjecture would give an example of a surface having nonrational Seshadri constant. Furthermore, it would also imply Nagata conjecturefor abc points in the projective plane. The latter result can be found for instancein [12, Prop. 5.2], but we give anyway a brief proof for the sake of completeness.
Proposition 3.3. If ε ( H, e ) = /√ abc then the Nagata conjecture holds for abc points in the plane.Proof. Let f ∶ P → P ( a, b, c ) be the morphism defined by ( x, y, z ) ↦ ( x a , y b , z c ) andlet Y r be the blowing-up of P at the r ∶ = abc points of f − ( e ) . Since R − is nef, also f ∗ R − = L − √ abc abc ∑ i = E i is nef on Y r . If we denote by X r the blowing-up of P at r points in very generalposition then Eff ( X r ) ⊆ Eff ( Y r ) by semicontinuity of the dimension of cohomology.Thus Nef ( X r ) ⊇ Nef ( Y r ) so that f ∗ R − is nef also on X r . (cid:3) Remark 3.4. If C is a very general smooth irreducible curve of positive genus g ,then the Ner´on-Severi group, over the rational numbers, of the symmetric productSym ( C ) has dimension two. In [10, Prop. 3.1] the authors show that if the Nagataconjecture is true and g ≥
9, then the effective cone of Sym ( C ) is open on one side.The equality ε ( H, e ) = /√ abc holds if and only if there does not exist a negativecurve in X ( a, b, c ) having class dπ ∗ H − mE , with d / m < √ abc . A partial result in thisdirection is given by [21, Thm. 5.4], where the authors show that if the negativecurve is expected, then d is bounded from above and this result allows them toconclude that there are no such curves on certain X ( a, b, c ) , like e.g. X ( , , ) .On the other hand we can also say that the equality ε ( H, e ) = /√ abc holds if andonly if there exists a sequence π ∗ d n H − m n E of classes of positive irreducible curvesin X ( a, b, c ) such that d n / m n → √ abc , that is these classes approach the ray R − from the inside of the light cone. In this direction observe that an intrinsic negative curve can appear as a positive curve in X ( a, b, c ) . We discuss this approach for X ( , , ) by producing many intrinsic ( − ) -curves which are positive curves onthe surface. We proceed using the fact that the Cox ring of X ( a, b, c ) is isomorphicto the extended saturated Rees algebra (see [18] and [4, Prop. 4.1.3.8]): R [ I ] sat ∶ = ⊕ m ∈ Z ( I m ∶ J ∞ ) t − m ⊆ R [ t ± ] , where R = K [ x, y, z ] is the Cox ring of the weighted projective plane, I is the idealof ( , , ) in Cox coordinates, J = ⟨ x, y, z ⟩ is the irrelevant ideal and I − m = R for any m ≥
0. Using this we can compute a minimal generating set consistingof homogeneous elements of given bounded multiplicity at ( , , ) . In the case of X ( , , ) , fixing the maximum of the multiplicity to be 30, we found 52 gener-ators. In the following table we display the degrees of these generators togetherwith the self-intersection of the corresponding intrinsic curve and its genus (whilethe self-intersection of the curve on X ( , , ) is d / abc − m ). d m C p a
36 1 0 039 1 0 040 1 0 083 2 -1 0109 3 -1 0110 3 -1 0113 3 -1 0139 4 -1 0140 4 -1 0143 4 -1 0208 6 -1 0209 6 -1 0210 6 -1 0 d m C p a
213 6 -2 0243 7 0 1309 9 -1 0310 9 -1 0312 9 -1 0313 9 -1 0378 11 -1 0379 11 -1 0380 11 -1 0413 12 -1 0481 14 -1 0482 14 -1 0483 14 -1 0 d m C p a
516 15 0 1549 16 -1 0550 16 -1 0551 16 -1 0585 17 -1 0652 19 -1 0653 19 -1 0686 20 0 1720 21 1 2721 21 -1 0755 22 -1 0789 23 0 1790 23 1 2 d m C p a
823 24 -1 0824 24 -1 0858 25 -1 0891 26 -1 0892 26 -1 0893 26 0 1893 26 3 3926 27 -1 0959 28 0 1960 28 0 1994 29 -1 01028 30 0 11029 30 -1 0
The slope d / m which best approximates √ ⋅ ⋅ ∼ . / = . ( − ) -curveof the above list is 891 / ∼ . Question 3.5.
Is it possible to construct an infinite family of intrinsic ( − ) -curvesappearing as positive curves in X ( , , ) , and whose slopes approach √ ⋅ ⋅ References [1] Florin Ambro and Atsushi Ito. Successive minima of line bundles.
Adv. Math.
J. Algebra
J. Pure Appl. Algebra
223 (11):4871–4887, 2019.[4] Ivan Arzhantsev, Ulrich Derenthal, J¨urgen Hausen, and Antonio Laface.
Cox rings . Cam-bridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge,2015.[5] Gabriele Balletti. Enumeration of Lattice Polytopes by Their Volume.
Discrete Comput.Geom. , 2020.[6] Thomas Bauer. Seshadri constants on algebraic surfaces.
Math. Ann.
313 (3):547–583, 1999.[7] Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Micha l Kapustka, Andreas Knutsen,Wioletta Syzdek, and Tomasz Szemberg. A primer on Seshadri constants. Interactions ofclassical and numerical algebraic geometry. Contemp. Math., vol. 496. Amer. Math. Soc.,Providence, RI., pages 33–70. 2009.[8] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The userlanguage.
Journal of Symbolic Computation
24 (3-4):235–265, 1997. Computational algebraand number theory (London, 1993).
N INTRINSIC NEGATIVE CURVES 15 [9] Ana-Maria Castravet, Antonio Laface, Jenia Tevelev, and Luca Ugaglia. Blown-up toric sur-faces with non-polyhedral effective cone. arXiv:2009.14298 , 2020.[10] Ciro Ciliberto and Alexis Kouvidakis. On the symmetric product of a curve with generalmoduli.
Geom. Dedicata
78 (3):327–343, 1999.[11] David A. Cox, John B. Little, and Henry K. Schenck.
Toric varieties . Graduate Studies inMathematics, vol. 124. American Mathematical Society, Providence, RI, 2011.[12] Steven Dale Cutkosky and Kazuhiko Kurano. Asymptotic regularity of powers of ideals ofpoints in a weighted projective plane.
Kyoto J. Math.
51 (1):25–45, 2011.[13] Carlos D’Andrea and Mart´ın Sombra. The Newton polygon of a rational plane curve.
Math.Comput. Sci.
Ast´erisque
Taiwanese J. Math.
21 (1):27–41, 2017.[16] Javier Gonz´alez Anaya, Jos´e Luis Gonz´alez, and Kalle Karu. Constructing non-Mori dreamspaces from negative curves.
J. Algebra arXiv:2008.04018 ,2020.[18] J¨urgen Hausen, Simon Keicher, and Antonio Laface. On blowing up the weighted projectiveplane.
Math. Z.
290 (3-4):1339–1358, 2018.[19] Atsushi Ito. Seshadri constants via toric degenerations.
J. Reine Angew. Math. arXiv:2101.02448 , 2021.[21] Kazuhiko Kurano and Naoyuki Matsuoka. On finite generation of symbolic Rees rings ofspace monomial curves and existence of negative curves.
J. Algebra
322 (9):3268–3290, 2009.[22] Alex K¨uronya, Victor Lozovanu, and Catriona Maclean. Convex bodies appearing as Ok-ounkov bodies of divisors.
Adv. Math.
229 (5):2622–2639, 2012.[23] Antonio Laface and Luca Ugaglia. On base loci of higher fundamental forms of toric varieties.
J. Pure Appl. Algebra
224 (12):106447, 18, 2020.[24] Michael Nakamaye. Seshadri constants and the geometry of surfaces.
J. Reine Angew. Math. k ∗ -action. Acta Math.
138 (1-2):43–81,1977.[26] Jenia Tevelev. Compactifications of subvarieties of tori.
Amer. J. Math.
129 (4):1087–1104,2007.
Departamento de Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on,Chile
Email address : [email protected] Dipartimento di Matematica e Informatica, Universit`a degli studi di Palermo, ViaArchirafi 34, 90123 Palermo, Italy
Email address ::