Bernstein-Sato Polynomials on Normal Toric Varieties
aa r X i v : . [ m a t h . AG ] A ug BERNSTEIN–SATO POLYNOMIALS ON NORMAL TORIC VARIETIES
JEN-CHIEH HSIAO AND LAURA FELICIA MATUSEVICHA
BSTRACT . We generalize the Bernstein–Sato polynomials of Budur, Mustat¸ˇa, and Saito to idealsin normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein–Sato polynomial to the jumping coefficients of the corresponding multiplier ideals. In order to provethe latter result, we obtain a new combinatorial description for the multiplier ideals of a monomialideal in a normal semigroup ring.
1. I
NTRODUCTION
Let f ∈ C [ x ] = C [ x , . . . , x n ] be a non-constant polynomial and let D = C [ x, ∂ x ] be the Weylalgebra. The Bernstein–Sato polynomial (or b -function) of f , introduced independently in [Ber72]and [SS72], is the monic polynomial b f ( s ) ∈ C [ s ] of smallest degree such that there exists P ( s ) ∈ D [ s ] = D ⊗ C C [ s ] satisfying the functional equation P ( s ) · f s +1 = b f ( s ) f s . It is well known that b f ( s ) = s + 1 if and only if the hypersurface defined by f is nonsingular. Moreover, the roots of b f ( s ) are related to the eigenvalues of the Milnor monodromy of f [Mal83], and the poles of thelocal zeta function associated to f [Igu00].Another important singularity invariant are the multiplier ideals J ( C n , α h f i ) associated to thehypersurface f in C n [Laz04]. They can be defined via an embedded log resolution of the pair ( C n , h f i ) . When the coefficient α varies, the multiplier ideal J ( C n , α h f i ) jumps. In fact, thesmallest jumping coefficient of ( C n , h f i ) (i.e. the log-canonical threshold of the hypersurfacedefined by f ) is the smallest root of b f ( − s ) [Yan83, Lic89, Kol97]. Generalizing this result,one of the main theorems in [ELSV04] states that if ξ is a jumping coefficient in (0 , of the pair ( C n , h f i ) , then it is a root of b f ( − s ) . Along the same lines, Budur, Mustat¸ˇa, and Saito extended thenotion of b -functions to the case of arbitrary ideals on smooth affine varieties [BMS06a]. Utilizingthe theory of V -filtrations from [Kas83] and [Mal83], they showed that the b -function of an ideal isindependent of the choice of generators. Furthermore, they generalized the connection establishedin [ELSV04] between the roots of their b -functions and the jumping coefficients of multiplier idealsto this more general setting. Theorem 1.1 ([BMS06a]) . Let I be an ideal on a smooth affine variety X over C . Then the log-canonical threshold of the pair ( X, I ) coincides with the smallest root α I of the b -function b I ( − s ) ,and any jumping coefficient of ( X, I ) in [ α I , α I + 1) is a root of b I ( − s ) . As the theory of multiplier ideals is generalized to the case of arbitrary ideals on normal vari-eties [dFH09], it is interesting to explore possible generalizations of Theorem 1.1. To this end, thefirst task is to seek for a suitable candidate that plays the role of the Weyl algebra. In the currentwork, we restrict to the case where the ambient variety X is an affine normal toric variety. There Mathematics Subject Classification.
Primary: 14F10; Secondary: 14M25, 14F18, 14B05.JCH was partially supported by MOST grant 105-2115-M-006-015-MY2.LFM was partially supported by NSF grant DMS 1500832. exists an explicit combinatorial description of Grothendieck’s ring of differential operators D X onthe toric variety X [Mus87, Jon94, ST01], serving as an analog of Weyl algebra in the case of X = C n . We apply this description of D X and the results in [BMS06a, BMS06b] to construct the b -function b I ( s ) for an ideal I on the toric variety X . To state our results more explicitly, let us setup some notations that will be used throughout this paper. Notation 1.2.
1. Throughout this article, we work over an algebraically closed field k of char-acteristic instead of the complex number field C . Our results are valid in this setting. Thecoordinate ring of the affine normal toric variety X is represented by the semigroup ring k [ N A ] generated by the columns a , . . . , a d of a rank d matrix A ∈ Z d × m . We assume that Z A = Z d , the cone C = R ≥ A in R d over A is strongly convex, and the semigroup N A isnormal, meaning that C ∩ Z d = N A . The faces of C are denoted by greek letters σ, τ etc.This may refer to an index set, to a collection of columns of A , or to the actual face of thecone. If σ is a facet of C , define its primitive integral support function F σ : R d → R suchthat(a) F σ ( Z A ) = Z ,(b) F σ ( a i ) ≥ for i = 1 , . . . , n ,(c) F σ ( a i ) = 0 for a i ∈ σ .Define the linear map F : R d → R F by F ( p ) = ( F σ ( p )) σ ∈ F , where F is the collection offacets of C .2. We consider k [ N A ] as a subring of the Laurent polynomial ring k [ y ± , . . . , y ± d ] . Then thering of differential operators D A on the toric variety X can be represented as a subring ofthe linear partial differential operators on ∂ y , . . . , ∂ y d with Laurent polynomial coefficients, D A = M u ∈ Z d y u { f ( θ ) ∈ k [ θ , . . . , θ d ] | f vanishes on N A r ( − u + N A ) } , where θ , . . . , θ d are the commuting operators y ∂ y , . . . , y d ∂ y d .3. The linear map F : R d → R F induces a ring homomorphism k [ N A ] → k [ N F ] via k X j =1 λ j y β j k X j =1 λ j x F ( β j ) , where β , . . . , β j ∈ N A and ( x σ ) σ ∈ F denote algebraically independent variables, so that k [ N F ] = k [ x σ | σ ∈ F ] . For an ideal I in k [ N A ] ⊂ k [ y ± , . . . , y ± d ] , we abuse notation todenote F ( I ) = (X λ j x F ( β j ) ∈ k [ N F ] (cid:12)(cid:12)(cid:12) X j λ j y β j ∈ I ) . Note that F ( I ) is an ideal in the semigroup ring k [ F ( N A )] . The ideal in k [ N F ] generatedby F ( I ) is denoted by J = h F ( I ) i .The first goal of this paper is to extend the notion of b -functions to the toric setting. We use the samedefinition as in [BMS06a] except that the Weyl algebra D is replaced by the ring of differentialoperators D A on X . Analyzing the combinatorics of the toric algebras, we obtain the followingresult. Theorem (Theorem 3.4) . The b -function b I ( s ) of an ideal I on the toric variety X coincides withthe b -function b J ( s ) of the ideal J = h F ( I ) i in the polynomial ring k [ N F ] . ERNSTEIN–SATO POLYNOMIALS ON NORMAL TORIC VARIETIES 3
In particular, avoiding the study of the V -filtration in the toric case, we can still conclude that thetoric b -function b I ( s ) is independent of the choices of generators of I .To generalize Theorem 1.1 to the toric setting, we further restrict to the case where I is a monomialideal (i.e. an ideal defining a torus-invariant subvariety of X ). In this case, the multiplier ideals ofthe pair ( X, I ) were described in [Bli04], generalizing the work in [How01] on the multiplier idealsof monomial ideals in polynomial algebras. Using these formulas, we obtain a new expression ofthe multiplier ideals of monomial ideals on toric varieties using the map F (Proposition 4.3), J ( X, αI ) = h y v ∈ k [ N A ] | F ( v ) + e ∈ int( αP J ) i , where e ∈ N F is such that e σ = 1 for all σ ∈ F and int( P J ) is the relative interior of the Newtonpolyhedron P J of the monomial ideal J in k [ N F ] . In particular, we obtain the identity, J ( X, αI ) = k [ N A ] ∩ J ( k F , αJ ) , which relates the jumping coefficients of the pair ( X, I ) to those of ( k F , J ) . Since the jumpingcoefficients of ( k F , J ) are related to the roots of b J ( − s ) by Theorem 1.1, our arguments give thefollowing theorem. Theorem (Theorem 4.4) . Let I be a monomial ideal on an affine normal toric variety X over analgebraically closed field k of characteristic . Then the log-canonical threshold of the pair ( X, I ) coincides with the smallest root α I of the b -function b I ( − s ) , and any jumping coefficients of ( X, I ) in [ α I , α I + 1) are roots of b I ( − s ) .
2. B
ERNSTEIN –S ATO P OLYNOMIALS OF B UDUR , M
USTAT ¸ ˘ A , AND S AITO
We recall relevant background on Bernstein–Sato polynomials for ideals from the work of Budur,Mustat¸˘a and Saito [BMS06a]. Here we concentrate on the case of subvarieties of affine spaces, butthe definition works for subvarieties of a smooth variety.Let I be an ideal in k [ x , . . . , x n ] , generated by f , . . . , f r . Denote by D the Weyl algebra on x , . . . , x n . If s , . . . , s r are indeterminates, consider k [ x , . . . , x n ][ r Y i =1 f − i , s , . . . , s r ] r Y i =1 f s i i . This is a D [ s ij ] -module, where s ij = s i t − i t j , and the action of the operator t i is given by t i ( s j ) = s j + δ ij (the Kronecker delta). Definition 2.1.
The
Bernstein–Sato polynomial or b -function associated to f = ( f , . . . , f r ) isdefined to be the monic polynomial of the lowest degree in s = P ri =1 s i satisfying a relation of theform b f ( s ) r Y i =1 f s i i = r X k =1 P k t k r Y i =1 f s i i , (2.1)where P , . . . , P r ∈ D [ s ij | i, j ∈ { , . . . , r } ] .In [BMS06a, Section 2], Budur, Mustat¸˘a and Saito show that:1. b f ( s ) is a nonzero polynomial,2. b f ( s ) is independent of the generating set of I . JEN-CHIEH HSIAO AND LAURA FELICIA MATUSEVICH
An alternative way to define the Bernstein–Sato polynomial.
In [BMS06a, Section 2.10]Budur, Mustat¸˘a and Saito give the following way to compute the b -function.For c = ( c , . . . , c r ) ∈ Z r , define nsupp( c ) = { i | c i < } . The Bernstein–Sato polyno-mial b f ( s ) is the monic polynomial of the smallest degree such that b f ( s ) Q ri =1 f s i i belongs tothe D [ s , . . . , s r ] -submodule generated by Y i ∈ nsupp( c ) (cid:18) s i − c i (cid:19) · r Y i =1 f s i + c i i (2.2)where c = ( c , . . . , c r ) runs over the elements of Z r such that P ri =1 c i = 1 . Here s = P ri =1 s i and (cid:0) s i m (cid:1) = s i ( s i − · · · ( s i − m + 1) /m ! .The advantage of this approach is that one now works in the ring D [ s , . . . , s r ] , and need no longerconsider the operators s ij or t k .2.2. A monomial-ideal-specific way to define the Bernstein–Sato polynomial.
The definitiongiven in the previous subsection can be further specialized to the case of monomial ideals.Let a be a monomial ideal in k [ x , . . . , x n ] whose minimal monomial generators are x α , . . . , x α r .Set ℓ α ( s , . . . , s r ) = s α + · · · + s r α r , and ℓ i ( s ) to be the i th coordinate of this vector. For v ∈ Z r ,let psupp( v ) = { i | v i > } .For c = ( c , . . . , c r ) ∈ Z r such that P ri =1 c i = 1 , set g c ( s , . . . , s r ) = Y j ∈ nsupp( c ) (cid:18) s j − c j (cid:19) · Y i ∈ psupp( ℓ α ( c )) (cid:18) ℓ i ( s , . . . , s r ) + ℓ i ( c ) ℓ i ( c ) (cid:19) , and define I a to be the ideal of Q [ s , . . . , s r ] generated by the polynomials g c .Then [BMS06a, Proposition 4.2] asserts that b f ( s ) is the monic polynomial of smallest degree suchthat b f ( P ri =1 s i ) belongs to the ideal I a .3. B ERNSTEIN –S ATO P OLYNOMIALS ON T ORIC V ARIETIES
The goal of this section is to extend Definition 2.1 to the toric setting. We first recall some back-ground on the ring of differential operators of a toric variety.3.1.
Rings of differential operators on toric varieties.
For a commutative algebra R over acommutative ring k , Grothendieck’s ring of k -linear differential operators D k ( R ) of R [Gro67] isthe R -subalgebra of Hom k ( R, R ) defined by D k ( R ) = [ i ∈ N ∪{ } D i , where D = R and D i = { f ∈ Hom k ( R, R ) | f r − rf ∈ D i − for all r ∈ R } for i ≥ . When R is the polynomial ring of n variables over an algebraically closed field k of characteristic , thering D k ( R ) coincides with the n th Weyl algebra. Moreover, it is well known that if R is regularover k , then D k ( R ) is the R -algebra generated by the k -derivations of R . The most influentialapplications of D -module theory, and in particular, the b -functions of [BMS06a], are built in thissetting. ERNSTEIN–SATO POLYNOMIALS ON NORMAL TORIC VARIETIES 5
When R is not regular, the ring D k ( R ) is not well-behaved [BGG72]. Nonetheless, when R = k [ N A ] is a toric algebra over an algebraically closed field k of characteristic , there exists anexplicit combinatorial description of D k ( k [ N A ]) . This expression of D k ( k [ N A ]) was obtainedindependently in [Mus87] and [Jon94] when N A is normal, and was further extended to the notnecessarily normal case in [ST01]. We denote the ring of differential operators of the toric algebra k [ N A ] by D A = D k ( k [ N A ]) . One way to understand D A is to identify k [ N A ] as a subring of the Laurent polynomial ring k [ y ± , . . . , y ± d ] and study the k -linear differential operators of k [ y ± , . . . , y ± d ] . It is known thatif S is a multiplicatively closed subset of a k -algebra R , then D k ( S − R ) ∼ = S − R ⊗ R D k ( R ) . Itfollows that D k ( k [ y ± , . . . , y ± d ]) = k [ y ± , . . . , y ± d ] h ∂ y , . . . , ∂ y d i , is a localization of the n th Weyl algebra. Moreover, one can realize D A = { δ ∈ D k ( k [ y ± , . . . , y ± d ]) | δ ( k [ N A ]) ⊆ k [ N A ] } as a subring of D k ( k [ y ± , . . . , y ± d ]) . In particular, we have the Z d -graded expression D A = M u ∈ Z d y u { f ( θ ) ∈ k [ θ , . . . , θ d ] | f vanishes on N A r ( − u + N A ) } (3.1)where θ , . . . , θ d are the commuting operators y i ∂ y i , for i = 1 , . . . , d . Since we are only concernedwith the case when N A is normal, for a given u ∈ Z d , the ideal { f ( θ ) ∈ k [ θ , . . . , θ d ] | f vanishes on N A r ( − u + N A ) } is the principal ideal in C [ θ , . . . , θ d ] generated by Y F σ ( u ) > F σ ( u ) − Y j =0 ( F σ ( θ , . . . , θ d ) − j ) , (3.2)where the product runs through all facets σ (with F σ ( u ) > ) of the cone C associated to N A , and F σ is the primitive integral support function of σ as defined in Notation 1.2.1.We are now ready to define the b -function for an ideal I in k [ N A ] . To make the exposition moretransparent, we first treat the case when I is an monomial ideal.3.2. Monomial ideals in semigroup rings.
Let a be a monomial ideal in k [ N A ] whose minimalmonomial generators are Laurent monomials y β , . . . , y β r . For f = ( y β , . . . , y β r ) , we define theBernstein–Sato polynomial b f ( s ) exactly as in Definition 2.1, except that the Weyl algebra D isreplaced by the ring of differential operators D A on k [ N A ] .We note that the reduction in Subsection 2.1 goes unchanged when we switch to the setting ofsemigroup rings, since that reduction is concerned only with the operators s ij and t k , and does notdepend at all on the ambient differential operators.Our task becomes to translate the situation from Subsection 2.2 to the new setting with monomialideals in semigroup rings, rather than monomial ideals in polynomial rings.Let c ∈ Z r whose coordinates sum to , and for β = ( β , . . . , β r ) ∈ ( Z d ) r , set ℓ β ( s , . . . , s r ) = s β + · · · + s r β r . JEN-CHIEH HSIAO AND LAURA FELICIA MATUSEVICH
In order to find the Bernstein–Sato polynomial, we need to apply an operator in D A [ s , . . . , s r ] to Q i ∈ nsupp( c ) (cid:0) s − c i (cid:1) · Q ri =1 ( y β i ) s i + c i in such a way that the outcome is a multiple of y P ri =1 s i β i = y ℓ β ( s ,...,s r ) .If we apply the operator from (3.2) with u = ℓ β ( c ) to Q i ∈ nsupp( c ) (cid:0) s − c i (cid:1) · Q ri =1 ( y β i ) s i + c i we obtain Y i ∈ nsupp( c ) (cid:18) s − c i (cid:19) Y F σ ( ℓ ( c )) > F σ ( ℓ ( c )) − Y j =0 ( F σ ( ℓ β ( s , . . . , s r ) + ℓ β ( c )) − j ) · y ℓ ( s ,...,s r ) . Proposition 3.1.
We use the notation introduced above. The Bernstein–Sato polynomial b f ( s ) for f = ( y β , . . . , y β r ) considered as a monomial ideal in k [ N A ] is the monic polynomial of smallestdegree such that b f ( s + · · · + s r ) belongs to the ideal generated by Y i ∈ nsupp( c ) (cid:18) s − c i (cid:19) Y F σ ( ℓ β ( c )) > (cid:18) F σ (cid:0) ℓ β ( s , . . . , s r ) + ℓ β ( c ) (cid:1) F σ ( ℓ β ( c )) (cid:19) (3.3) where c ∈ Z r has coordinate sum .Proof. It remains to be shown that, for any other operator (in D A [ s , . . . , s r ] ) which applied to Q i ∈ nsupp( c ) (cid:0) s − c i (cid:1) · Q ri =1 ( y β i ) s i + c i yields a multiple of y ℓ ( s ,...,s r ) , this multiple belongs to the idealgenerated by (3.3).This follows from the description of D A given by (3.1) and (3.2). (cid:3) Recall that F is the collection of facets of the cone C of N A , and that the map F : R d → R F defined by F ( p ) = ( F σ ( p )) σ ∈ F from Notation 1.2.1 induces an inclusion k [ N A ] → k [ N F ] . Proposition 3.2.
Denote by F ( f ) the sequence of monomials ( x F ( β ) , . . . , x F ( β r ) ) in k [ N F ] . TheBernstein–Sato polynomial b F ( f ) ( s ) of F ( f ) in the polynomial ring k [ N F ] coincides with theBernstein–Sato polynomial b f ( s ) of f = ( x β , . . . , x β r ) in k [ N A ] . In particular, the latter polyno-mial b f ( s ) is nonzero, and its roots can be computed using the combinatorial description [BMS06b,Theorem 1.1] applied to h x F ( β ) , . . . , x F ( β r ) i ⊆ k [ N F ] .Proof. First we observe that the minimal monomial generators of h x F ( β ) , . . . , x F ( β r ) i ⊆ k [ N F ] are the monomials x F ( β ) , . . . , x F ( β r ) . To see this, note that x F ( β i ) divides x F ( β j ) is equivalent to F σ ( β i ) ≤ F σ ( β j ) for all facets σ of the cone C . This implies that β j − β i lies in the cone C . Since β j − β i ∈ Z r and N A is normal, we see that β j − β i ∈ N A , and therefore y β i divides y β j in k [ N A ] .Let α i = F ( β i ) for i = 1 , . . . , r , and ℓ α ( s , . . . , s r ) = s α + · · · + s r α r , and denote by ℓ i ( s , . . . , s r ) the i th coordinate of the vector ℓ α ( s , . . . , s r ) . By [BMS06a, Proposition 4.2] theBernstein–Sato polynomial of h x F ( β ) , . . . , x F ( β r ) i is the monic polynomial p of smallest degreesuch that p ( s + · · · + s r ) lies in the ideal generated by Y j ∈ nsupp( c ) (cid:18) s j − c j (cid:19) Y i ∈ psupp( ℓ α ( c )) (cid:18) ℓ i ( s , . . . , s r ) + ℓ i ( c ) ℓ i ( c ) (cid:19) (3.4)for c ∈ Z r with coordinate sum . But note that by construction ℓ α = F ◦ ℓ β , so that the gen-erators (3.4) coincide exactly with the generators (3.3). As a reality check, note that both setsof generators belong to the ring k [ s , . . . , s r ] , which does not depend on the ambient rings of themonomial ideals involved. (cid:3) ERNSTEIN–SATO POLYNOMIALS ON NORMAL TORIC VARIETIES 7
The general case.
Again, recall that we have a linear map F : R d → R F with F ( N A ) ⊆ N F .We construct a ring homomorphism k [ N A ] → k [ N F ] via k X j =1 λ j y β j k X j =1 λ j x F ( β j ) , (3.5)where β , . . . , β j ∈ N A . That this is a homomorphism follows from linearity of F , as x F ( β + β ′ ) = x F ( β )+ F ( β ′ ) = x F ( β ) x F ( β ′ ) . We abuse notation and denote the homomorphism (3.5) by F .If I ⊂ k [ N A ] is an ideal, then its image F ( I ) is not an ideal in k [ N F ] ; we denote by h F ( I ) i theideal in k [ N F ] generated by F ( I ) . Lemma 3.3.
Let I ⊂ k [ N A ] be an ideal generated by g , . . . , g r ∈ k [ N A ] . Then F ( g ) , . . . , F ( g r ) generate h F ( I ) i .Proof. Since g , . . . , g r generate I , and F is a ring homomorphism, any element of F ( I ) is obtainedas a combination of F ( g ) , . . . , F ( g r ) with coefficients in k [ N F ] . This implies that the polynomials F ( g ) , . . . , F ( g r ) generate h F ( I ) i . (cid:3) We apply Definition 2.1 to a sequence of polynomials in k [ N A ] , using the ring of differentialoperators D A instead of D . The following is the main result of this section. Theorem 3.4.
Let I ⊂ k [ N A ] be an ideal, and let g , . . . , g r be generators for I . The Bernstein–Sato polynomial of g = ( g , . . . , g r ) in k [ N A ] equals the Bernstein–Sato polynomial of f =( f , . . . , f r ) = F ( g ) = ( F ( g ) , . . . , F ( g r )) in k [ N F ] . Consequently, this Bernstein–Sato poly-nomial is nonzero, and depends only on I , not on the particular set of generators chosen.Proof. We know that the Bernstein–Sato polynomial b g ( s ) is the monic polynomial of smallestdegree such that, for s = s + · · · + s r , b g ( s ) Q ri =1 g s i i belongs to the D A [ s , . . . , s r ] -submodulegenerated by Y i ∈ nsupp( c ) (cid:18) s i − c i (cid:19) · r Y i =1 g s i + c i i , for ( c , . . . , c r ) ∈ Z r with r X i =1 c i = 1 , while b f ( s ) is is the monic polynomial of smallest degree such that, b f ( s ) Q ri =1 f s i i belongs to the D [ s , . . . , s r ] -submodule generated by Y i ∈ nsupp( c ) (cid:18) s i − c i (cid:19) · r Y i =1 f s i + c i i , for ( c , . . . , c r ) ∈ Z r with r X i =1 c i = 1 . Here D is the ring of differential operators on k [ N F ] , and D A is the ring of differential operatorson k [ N A ] .Fix c ∈ Z r such that P c i = 1 , and let P ∈ D [ s , . . . , s r ] such that P (cid:2) Q i ∈ nsupp( c ) (cid:0) s i − c i (cid:1) · Q ri =1 f s i + c i i (cid:3) is a polynomial in s , . . . , s r times Q ri =1 f s i i . Applying the description given by (3.1)and (3.2) to D (instead of D A ), we see that we can write P as a finite sum P = X u ∈ Z F q u ( s , . . . , s r ) x u (cid:20) Y u σ < − u σ − Y j =0 (cid:0) ( θ x ) σ − j (cid:1)(cid:21) p u ( θ x ) , where the p u are polynomials in | F | indeterminates with coefficients in k . JEN-CHIEH HSIAO AND LAURA FELICIA MATUSEVICH
Note that, by construction, the element Q i ∈ nsupp( c ) (cid:0) s i − c i (cid:1) · Q ri =1 f s i + c i i is F ( N A ) -graded. Since P (cid:2) Q i ∈ nsupp( c ) (cid:0) s i − c i (cid:1) · Q ri =1 f s i + c i i (cid:3) is a multiple of Q ri =1 f s i i , we may assume that the operator P is F ( N A ) -graded as well. In other words, we may assume that u ∈ F ( Z d ) if q u p u = 0 .Thus we rewrite P = X u = F ( v ) ∈ F ( Z d ) q u ( s , . . . , s r ) x u (cid:20) Y u σ < − u σ − Y j =0 (cid:0) ( θ x ) σ − j (cid:1)(cid:21) p u ( θ x ) , and if we denote ˆ P = X v ∈ Z d q F ( v ) ( s , . . . , s r ) y v (cid:20) Y F σ ( v ) < − F σ ( v ) − Y j =0 (cid:0) F σ ( θ y ) − j (cid:1)(cid:21) p F ( v ) (( F σ ( θ y )) σ ∈ F ) , then ˆ P is an element of D A [ s , . . . , s r ] , and ˆ P applied to Q i ∈ nsupp( c ) (cid:0) s i − c i (cid:1) · Q ri =1 g s i + c i i is a poly-nomial of s , . . . , s r times Q ri =1 g s i i , and this is the same polynomial that we obtain when we apply P to Q i ∈ nsupp( c ) (cid:0) s i − c i (cid:1) · Q ri =1 f s i + c i i (and divide by Q f s i i ).Conversely, if we apply an element of D A [ s , . . . , s r ] to Q i ∈ nsupp( c ) (cid:0) s i − c i (cid:1) · Q ri =1 g s i + c i i and obtaina polynomial in s , . . . , s r times Q ri =1 g s i i , then we can obtain an element of D [ s , . . . , s r ] whichapplied to Q i ∈ nsupp( c ) (cid:0) s i − c i (cid:1) · Q ri =1 f s i + c i i gives exactly the same polynomial in s , . . . , s r times Q ri =1 f s i i . To do this, we write the element of D A [ s , . . . , s r ] as a sum of terms of the form q v ( s , . . . , s r ) y v (cid:20) Y F σ ( v ) < − F σ ( v ) − Y j =0 ( F σ ( θ y ) − j ) (cid:21) p v ( θ y ) , then express (non uniquely) the polynomial p v as sums of powers of the linear forms F σ ( θ y ) , andthen apply F in the obvious way. Nonuniqueness comes because we need to choose d linearlyindependent forms F σ in order to get k -algebra generators of k [ θ y ] ; however, since the image of F as a linear map is d -dimensional, the choice does not affect the image. (cid:3)
4. M
ULTIPLIER I DEALS ON N ORMAL T ORIC V ARIETIES
We recall basic definitions of multiplier ideals that can be found in Lazarsfeld’s text, Positivity inAlgebraic Geometry II [Laz04].Let X be a smooth variety over an algebraically closed field k of characteristic . Let I ⊆ O X be an ideal sheaf, and α > a rational number. Fix a log resolution µ : X ′ → X of I with I · O X ′ = O X ′ ( − E ) . The multiplier ideal of the pair ( X, α I ) is defined as J ( X, α I ) = µ ∗ O X ′ ( K X ′ /X − ⌊ α · E ⌋ ) , where K X ′ /X = K X ′ − µ ∗ K X is the relative canonical divisor of µ , which is an effective divisorsupported on the exceptional locus of µ whose local equation is given by the determinant of thederivative dµ . The definition does not depend on the choice of log resolution. One of the importantfeatures of multiplier ideals is that J ( X, α I ) measures the singularity of the pair ( X, α I ) : asmaller multiplier ideal corresponds to a worse singularity.Notice that J ( X, α I ) becomes smaller as α increases. The jumping coefficients of ( X, α I ) arethe positive real numbers < α < α < . . . such that J ( X, α j I ) = J ( X, α I ) = J ( X, α j +1 I ) for α j ≤ α < α j +1 ( j ≥ where α = 0 . When X is affine and I is the ideal in the coordinate ring ERNSTEIN–SATO POLYNOMIALS ON NORMAL TORIC VARIETIES 9 k [ X ] corresponding the the sheaf I , Budur, Mustat¸˘a and Saito proved that jumping coefficients of ( X, αI ) in [ α f , α f + 1) are roots of b f ( − s ) , where f = ( f , . . . , f r ) is a set of generators for I , b f ( s ) is the Bernstein–Sato polynomial of I , and α f is the smallest root of b f ( − s ) . One of ourgoals is to generalize this correspondence between b -function roots and jumping coefficients tomonomial ideals on affine normal toric varieties.In the special case that I is a monomial ideal in the polynomial ring k [ x , . . . , x m ] , Howald gavethe following combinatorial formula for multiplier ideal of the pair ( A m , αI ) , J ( A m , αI ) = h x v | v + e ∈ int ( αP I ) i , where P I is the Newton polyhedron of I and e is the vector (1 , , . . . , in N m . In particular,a rational number α > is a jumping coefficient of ( A m , αI ) if and only if the boundary of − e + int ( αP I ) contains a lattice point in N m .The notion of multiplier ideal can be generalized to the case that I is an ideal sheaf on a normalvariety X over k as follows. Let ∆ be an effective Q -divisor such that K X + ∆ is Q -Cartier. Such ∆ is called a boundary divisor. Let µ : X ′ → X be a log resolution of the triple ( X, ∆ , I ) and α > be a rational number. Suppose that I · O X ′ = O X ′ ( − E ) . Then one can define the multiplierideal J ( X, ∆ , α I ) associated to the triple ( X, ∆ , α I ) as J ( X, ∆ , α I ) = µ ∗ O X ′ ( K X ′ − ⌊ µ ∗ ( K X + ∆) + αE ⌋ ) . Again, this definition is not dependent on the choice of µ . De Fernex and Hacon [dFH09] havegiven a definition of J ( X, αZ ) for non- Q -Gorenstein X without using the boundary divisor ∆ .They showed that there exists a boundary divisor ∆ such that the multiplier ideal J ( X, ∆ , αZ ) coincides with their multiplier ideal J ( X, αZ ) and that J ( X, αZ ) is the unique maximal elementof the set {J ( X, ∆ , αZ ) | ∆ is a boundary divisor } . A New expression for multiplier ideals on toric varieties.
Let us explain how one can usethe map F to understand the multiplier ideals of De Fernex and Hacon in the case of monomialideals on normal toric varieties.Let k [ N A ] ⊂ k [ y ± , . . . , y ± d ] be a normal semigroup ring in our setting and let X = Spec( k [ N A ]) .Each facet σ ∈ F corresponds to a torus invariant prime Weil divisor D σ on X . The canonicalclass of X is represented by the torus invariant canonical divisor K X = − P σ ∈ F D σ . A boundarydivisor ∆ is an effective Q -divisor such that K X + ∆ is Q -Cartier, which means there exist l ∈ Z and u ∈ Z d such that l ( K X + ∆) = div( y u ) . Denote w ∆ = ul . Notice that ∆ = X σ ∈ F (1 + F σ ( w ∆ )) D σ , so the effectivity of ∆ is equivalent to the condition that F σ ( w ∆ ) ≥ − for all σ ∈ F . Conversely,each w ∈ Q d gives rise to a boundary divisor ∆ w on X by the same formula ∆ w = X σ ∈ F (1 + F σ ( w )) D σ . Let I be a monomial ideal in k [ N A ] and let α > be a rational number. Blickle [Bli04] showed thatthe multiplier ideal J ( X, ∆ , αI ) = h y v ∈ k [ N A ] | v − w ∆ ∈ int( αP I ) i , generalizing Howald’s description of multiplier ideals [How01] in the case of monomial ideals in polynomial rings. Becautioned about the mistake of the sign of w ∆ in Blickle’s original description. For w ∈ Q d , denote Ω w,αI = [ w + int( αP I )] and Ω αI = [ w ∈ Q d : F σ ( w ) ≥− ∀ σ ∈ F Ω w,αI . (4.1)Then the multiplier ideal of De Fernex and Hacon is J ( X, αI ) = h y v ∈ k [ N A ] | v ∈ Ω αI i . (4.2)We claim that J ( X, αI ) can be computed using an analog of Howald’s formula. Recall that thelinear map F : R d → R F induces ring homomorphisms k [ N A ] ∼ −→ k [ F ( N A )] → k [ N F ] . By abusing notation, denote F ( I ) = k [ F ( N A )] · h x F ( v ) | y v ∈ I i (4.3)the ideal in k [ F ( N A )] obtained from the semigroup isomorphism F : N A ∼ −→ F ( N A ) . The mono-mial ideal in k [ N F ] generated by F ( I ) is denoted by J = k [ N F ] · h F ( I ) i . (4.4)Let e be the element in R F such that e σ = 1 for all σ ∈ F . Notice that e may not be in F ( N A ) and that e ∈ F ( N A ) if and only if X is Gorenstein. Even if one extends to rational coefficients,the element e may not be in F ( Q ⊗ N A ) = F ( Q d ) . The condition e ∈ F ( Q d ) holds exactly when X is Q -Gorenstein, namely K X is a Q -Cartier divisor. This is the case when X is Q -factorial (or,equivalently, the semigroup N A is simplicial).In the case where e ∈ F ( Q d ) , it is clear that Ω αI = Ω F − ( − e ) ,αI , so w = F − ( − e ) correspondsto the ∆ of De Fernex and Hacon. In fact, the divisor ∆ F − ( − e ) = 0 coincides with the canonicalchoice of boundary divisor for the pair ( X, αI ) in the Q -Gorenstein case. Moreover, we have thefollowing analog of Howald’s formula for J ( X, αI ) . Proposition 4.1. If X is Q -Gorenstein, then J ( X, αI ) = h y v ∈ k [ N A ] | F ( v ) + e ∈ int( αP F ( I ) ) i . Proof. J ( X, αI ) = h y v ∈ k [ N A ] | v ∈ Ω αI i = h y v ∈ k [ N A ] | v ∈ Ω F − ( − e ) ,αI i = h y v ∈ k [ N A ] | F ( v ) + e ∈ int( αF ( P I )) i = h y v ∈ k [ N A ] | F ( v ) + e ∈ int( αP F ( I ) ) i . (4.5)Since P I = conv hS v : y v ∈ I ( v + C ) i is the convex hull in R d of the set S y v ∈ I ( v + C ) , we see that F ( P I ) = conv hS v : y v ∈ I ( F ( v ) + F ( C )) i is exactly the Newton polyhedron P F ( I ) in F ( R d ) of theideal F ( I ) in the semigroup ring k [ F ( N A )] . The last expression in (4.5) is analogous to Howald’sformula. (cid:3) ERNSTEIN–SATO POLYNOMIALS ON NORMAL TORIC VARIETIES 11
Notations for Newton Polytopes.
For a monomial ideal I in k [ N A ] ⊂ k [ y ± , . . . , y ± d ] , denote P I the Newton polyhedron of I , which by definition is the convex hull of S v : y v ∈ I ( v + N A ) in R d .The relative interior of P I is denoted by int P I .Recall from (4.3) that F ( I ) denotes a monomial ideal in k [ N A ] . The Newton polyhedron of F ( I ) is denoted by P F ( I ) ; this is the convex hull of { F ( v ) | y v ∈ I } in F ( R d ) . We have P F ( I ) = conv " [ v : y v ∈ I ( F ( v ) + F ( N A )) . Recall also from (4.4) that J = k [ N F ] · h F ( I ) i denotes the monomial ideal in k [ N F ] generated by F ( I ) . The Newton polyhedron of J in R F is P J = conv " [ v : y v ∈ I (cid:0) F ( v ) + N F (cid:1) . Note that if y β , . . . , y β r is the set of minimal monomial generators of I , then { β , . . . , β r } (re-spectively, { F ( β ) , . . . , F ( β r ) } ) is exactly the set of vertices of the Newton polyhedron P I (re-spectively, P F ( I ) or P J ).For general X , we give a description of J ( X, αI ) using the Newton polyhedron of the ideal J = k [ N F ] · h F ( I ) i . We first need to compare the relative interiors of αP F ( I ) and αP J . Lemma 4.2.
For general N A , the relative interiors of P F ( I ) and P J satisfy int (cid:0) αP F ( I ) (cid:1) = F ( C ) ∩ int ( αP J ) . Proof.
A point u lies in int( P F ( I ) ) if and only if u − v ∈ int( F ( C )) for some v on a bounded faceof P F ( I ) . Similarly, a point u lies in int( P J ) if and only if u − v ∈ int R F ≥ for some v on a boundedface of P J . Since F ( I ) and J have the same minimal monomial generators, the bounded faces of P F ( I ) and P J coincide. Therefore, it suffices to show that int( F ( C )) = F ( C ) ∩ int( R F ≥ ) . But for p ∈ C , F ( p ) ∈ int F ( C ) if and only if p is not contained in any facet σ ∈ F , which meansexactly F σ ( p ) > . (cid:3) Proposition 4.3.
The multiplier ideal J ( X, αI ) = h y v ∈ k [ N A ] | F ( v ) + e ∈ int( αP J ) i . Proof. If X is Q -Gorenstein, the statement follows from Proposition 4.1 and Lemma 4.2.In general, by (4.1) and (4.2) it suffices to show that F (Ω αI ∩ N A ) = ( − e + int ( αP J )) ∩ F ( N A ) . For the containment F (Ω αI ∩ N A ) ⊆ ( − e + int ( αP J )) ∩ F ( N A ) , it is enough to verify that F ( w + int( αP I )) ⊆ [ − e + int ( αP J )] for any w ∈ Q d satisfying F σ ( w ) ≥ − for all σ ∈ F . Any such w satisfies F ( w ) + e ∈ R F ≥ , soby Lemma 4.2 F ( w + int( αP I )) = [ F ( w ) + int ( αF ( P I ))]= (cid:2) F ( w ) + int (cid:0) αP F ( I ) (cid:1)(cid:3) ⊆ [ F ( w ) + int ( αP J )]= [ − e + ( F ( w ) + e ) + int ( αP J )] ⊆ (cid:2) − e + R F ≥ + int ( αP J ) (cid:3) ⊆ [ − e + int ( αP J )] . For the other containment F (Ω αI ∩ N A ) ⊇ ( − e + int ( αP J )) ∩ F ( N A ) , let v ∈ N A be such that F ( v ) + e ∈ int( αP J ) . Then there exists u lying on a bounded face of αP I such that F ( v ) + e − F ( u ) ∈ int( R F ≥ ) In particular, F σ ( v − u ) > − for all σ ∈ F . Take any p ∈ int C and ǫ > small enough so that w := v − u − ǫp ∈ Q d and F σ ( v − u − ǫp ) > − . Then F ( v ) − F ( u ) − F ( w ) = F ( ǫp ) ∈ int C where F ( u ) lies on a bounded face of αP F ( I ) .Therefore, we have F ( v ) − F ( w ) ∈ int (cid:0) αP F ( I ) (cid:1) , and hence F ( v ) ∈ F (Ω w,αI ∩ N A ) ⊆ F (Ω αI ∩ N A ) as desired. (cid:3) Roots–Jumping coefficients correspondence on toric varieties.
Now, we combine the pre-vious observations to establish the following theorem.
Theorem 4.4.
Let N A be a normal semigroup, X = Spec( k [ N A ]) its associated affine toric varietyand let I be a monomial ideal on X . Suppose α I is the smallest root of b I ( − s ) where b I ( s ) is theBernstein–Sato polynomial of I in k [ N A ] . Then any jumping coefficients of the pair ( X, I ) in [ α I , α I + 1) are roots of b I ( − s ) . Moreover, the number α I is the smallest jumping coefficient (i.e.the log-canonical threshold) of ( X, I ) .Proof. By Proposition 3.2, the Berstein–Sato polynomial b I ( s ) of I coincides with the Bernstein–Sato polynomial of the monomial ideal J = k [ N F ] · h F ( I ) i . Thus by [BMS06a, Theorem 2], thejumping coefficients of the pair ( A F , J ) in [ α I , α I + 1) are roots of b I ( − s ) . By Howald’s formula,a number α is a jumping coefficient of ( A F , J ) exactly when the boundary of ( − e + αP J ) containsa lattice point in N F . Also, according to Proposition 4.3 a number α is a jumping coefficient of ( X, I ) exactly when the boundary of ( − e + αP J ) contains a lattice point in F ( N A ) . Therefore,the jumping coefficients of ( X, I ) are jumping coefficients of ( A F , J ) , and the first statement ofthis theorem follows.To prove α I is the log-canonical threshold of ( X, I ) , it suffices to show that the boundary of ( − e + α I P J ) intersects F ( N A ) . Since α I is the log-canonical threshold of ( A F , J ) , we have ∈ k [ N F ] = J ( A F , α I J ) , and hence F (0) = 0 ∈ ( − e + α I P J ) ∩ F ( N A ) . (cid:3) Example 4.5.
Let A = (cid:18) (cid:19) and I = h y y , y y i a monomial ideal in k [ N A ] . Inthis case, the semigroup ring k [ N A ] is simplicial and hence Q -Gorenstein. The linear mapping ERNSTEIN–SATO POLYNOMIALS ON NORMAL TORIC VARIETIES 13 F : R → R is represented by the matrix (cid:18) −
10 1 (cid:19) . The subsemigroup k [ F ( N A )] of k [ x , x ] is generated by x , x x , x x , x and the monomial ideal J = k [ x , x ] h x x , x x i . Using themethod of Budur, Mustat¸˘a and Saito as discussed in subsection 2.2, one can compute the Bernstein–Sato polynomial b J ( s ) = ( s + 1) (3 s + 2)(3 s + 4) . (We point out that b -function algorithms havebeen developed and implemented, see [BL10].) Moreover, by Proposition 3.2 we have b I ( s ) =( s + 1) (3 s + 2)(3 s + 4) as well. On the other hand, using Howald’s formula, one finds that , , are jumping coefficients of ( A , J ) , but only , are jumping coefficients of ( X, I ) according toProposition 4.3. A CKNOWLEDGEMENTS
The first author thanks Texas A&M University for the hospitality he enjoyed during his visit inJanuary 2016, when this work was initiated.R
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