Bernstein–Sato Polynomials on Normal Toric Varieties Jen-Chieh Hsiao

Bernstein–Sato Polynomials on Normal Toric Varieties

Jen-Chieh Hsiao & Laura Felicia Matusevich

Abstract. We generalize the Bernstein–Sato polynomials of Budur, Musta¸tˇa, and Saito to ideals in 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 multi- plier ideals. To prove the latter result, we obtain a new combinatorial description for the multiplier ideals of a monomial ideal in a normal semigroup ring.

1. Introduction

Let f ∈ C[x] = C[x¹, . . . , x_n] be a nonconstant polynomial, and let D = C[x, ∂^x] be the Weyl algebra. The Bernstein–Sato polynomial (or b-function) of f , in- troduced 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 ⊗_CC[s] satisfying the functional equation P (s)·f^s+1= b^f(s)f^s. It is well known that bf(s)= s +1 if and only if the hypersurface defined by f is nonsingular. Moreover, the roots of b_f(s)are conjecturally related to the eigenvalues of the Milnor monodromy of f [Mal83] and the poles of the local zeta function associated with f [Igu00].

Another important singularity invariant are the multiplier ideals J (Cⁿ, α⟨f ⟩) associated with the hypersurface defined by f in Cⁿ [Laz04]. They can be de- fined via an embedded log resolution of the pair (Cⁿ,⟨f ⟩). When the coefficient α varies, the multiplier ideal J (Cⁿ, α⟨f ⟩) jumps. In fact, the smallest jumping coefficient of (Cⁿ,⟨f ⟩) (i.e. the log-canonical threshold of the hypersurface de- fined by f ) is the smallest root of bf(−s) [Yan83; Lic89; Kol97]. Generalizing this result, one of the main theorems in [ELSV04] states that if ξ is a jumping coefficient in (0, 1] of the pair (Cⁿ,⟨f ⟩), then it is a root of bf(−s). Along the same lines, Budur, Musta¸tˇa, and Saito extended the notion of b-functions to the case of arbitrary ideals on smooth affine varieties [BMS06a]. Utilizing the theory of V -filtrations from [Kas83] and [Mal83], they showed that the b-function of an ideal is independent of the choice of generators. Furthermore, they generalized the connection established in [ELSV04] between the roots of their b-functions and the jumping coefficients of multiplier ideals to this more general setting.

Received August 11, 2016. Revision received September 18, 2017.

JCH was partially supported by Taiwan MOST grant 105-2115-M-006-015-MY2. LFM was partially supported by NSF grant DMS-1500832.

117

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 small- est 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 varieties [dFH09], it is interesting to explore possible generalizations of Theorem 1.1. To this end, the first task is to look for a suitable candidate that plays the role of the Weyl algebra. In the current work, we restrict to the case where the ambient variety X is an affine normal toric variety. There exists an explicit combinatorial description of Grothendieck’s ring of differential operators D_X on the toric variety X [Mus87; Jon94; ST01], serving as an analog of Weyl algebra in the case of X= Cⁿ. We apply this description of DX and the results in [BMS06a; BMS06b] to construct the b-function bI(s) for an ideal I on the toric variety X. To state our results more explicitly, let us set up some notations.

Notation 1.2. 1. Throughout this article, we work over an algebraically closed field k of characteristic 0 instead of the complex number field C. Our results are valid in this setting. The coordinate ring of the affine normal toric vari- ety X is represented by the semigroup ring k[NA] generated by the columns a₁, . . . , a_m of a rank d matrix A∈ Z^d×m. We assume thatZA = Z^d, the cone C = R≥0AinR^d over A is strongly convex, and the semigroupNA is normal, meaning that C ∩ Z^d = NA. The faces of C are denoted by greek letters σ , τ, and so on. This may refer to an index set, to a collection of columns of A, or to the actual face of the cone. If σ is a facet of C, then define its primitive integral support function F_σ : R^d → R such that

(a) Fσ(ZA) = Z,

(b) Fσ(a_i)≥ 0 for i = 1, . . . , n, and (c) Fσ(a_i)= 0 for aⁱ∈ σ .

Define the linear map F : R^d → R^F by F (p)= (F^σ(p))_σ∈F, where F is the collection of facets of C.

2. We consider k[NA] as a subring of the Laurent polynomial ring k[y₁^±1, . . . , y_d^±1]. Then the ring of differential operators D^A on the toric variety X can be represented as a subring of the linear partial differential operators on

∂_y1, . . . , ∂_yd with Laurent polynomial coefficients, D_A= !

u∈Z^d

y^u{f (θ) ∈ k[θ¹, . . . , θ_d] | f vanishes on NA \ (−u + NA)}, where θ₁, . . . , θ_d are the commuting operators y₁∂_y1, . . . , y_d∂_yd.

3. The linear map F : R^d → R^F induces a ring homomorphism k[NA] → k[N^F] via

"k j=1

λ_jy^βj *→

"k j=1

λ_jx^{F (βj)},

where β1, . . . , β_j ∈ NA, and (xσ)_σ∈F denote algebraically independent variables, so that k[N^F] = k[x^σ | σ ∈ F ]. For an ideal I in k[NA] ⊂

k[y₁^±1, . . . , y_d^±1], we abuse notation to denote F (I )=#"

λ_jx^{F (βj)} ∈ k[N^F]$

$$"

j

λ_jy^βj ∈ I

% .

Note that F (I ) is an ideal in the semigroup ringk[F (NA)]. The ideal in k[N^F] generated by F (I ) is denoted by J = ⟨F (I)⟩.

4. For a monomial ideal I ink[NA] ⊂ k[y₁^±1, . . . , y_d^±1], denote by P^I the Newton polyhedron of I , which by definition is the convex hull of &

v:y^v∈I(v+ NA) inR^d. The relative interior of P_I is denoted by int P_I.

The first goal of this paper is to extend the notion of b-functions to the toric set- ting. We use the same definition as in [BMS06a] except that the Weyl algebra D is replaced by the ring of differential operators D_A on X. Analyzing the combi- natorics of the toric algebras, we obtain the following result.

Theorem (Theorem 3.4). The b-function bI(s) of an ideal I on the toric variety X coincides with the b-function b_J(s) of the ideal J = ⟨F (I)⟩ in the polynomial ringk[N^F].

In particular, avoiding the study of the V -filtration in the toric case, we can still conclude that the toric b-function bI(s)is independent of the choice of generators of I .

To generalize Theorem 1.1 to the toric setting, we further restrict to the case where I is a monomial ideal (i.e. an ideal defining a torus-invariant subvariety of X). In this case, the multiplier ideals of the pair (X, I ) were described in [Bli04], generalizing the work in [How01] on the multiplier ideals of monomial ideals in polynomial algebras. Using these formulas, we obtain a new expression of the multiplier ideals of monomial ideals on toric varieties using the map F (Proposi- tion4.2),

J (X, αI ) = ⟨y^v∈ k[NA] | F (v) + e ∈ int(αP^J)⟩,

where e∈ N^F is such that eσ = 1 for all σ ∈ F , and int(P^J)is the relative interior of the Newton polyhedron PJ of the monomial ideal J in k[N^F]. In particular, we obtain the identity

J (X, αI ) = k[NA] ∩ J (k^F, αJ ),

which relates the jumping coefficients of the pair (X, I ) to those of (k^F, J ).

Since the jumping coefficients of (k^F, J ) are related to the roots of bJ(−s) by Theorem1.1, our arguments give the following theorem.

Theorem (Theorem 4.4). Let I be a monomial ideal on an affine normal toric variety X over an algebraically closed field k of characteristic 0. 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 all jumping coefficients of (X, I) in [α^I, α_I + 1) are roots of b_I(−s).

To end this introduction, we mention the work of Àlvarez-Montaner, Huneke, and Núñez-Betancourt [ÀHN16], where b-functions are studied in the context of direct summand rings.

2. Bernstein–Sato Polynomials of Budur, Musta¸t˘a, and Saito We recall relevant background on Bernstein–Sato polynomials for ideals from the work of Budur, Musta¸t˘a, and Saito [BMS06a]. Here we concentrate on the case of subvarieties of affine spaces, but the definition works for subvarieties of a smooth variety.

Let I be an ideal in k[x¹, . . . , x_n] generated by f¹, . . . , f_r. Denote by D the Weyl algebra on x1, . . . , x_n. If s1, . . . , s_r are indeterminates, then consider

k[x¹, . . . , x_n] '(^r

i=1

f_i⁻¹, s₁, . . . , s_r )(^r

i=1

f_i^si.

This is a D[s^ij]-module, where s^ij = sⁱt_i⁻¹t_j, and the action of the operator ti is given by t_i(s_j)= sj + δij (the Kronecker delta).

Definition 2.1. The Bernstein–Sato polynomial or b-function associated with f = (f¹, . . . , f_r) is defined to be the monic polynomial of lowest degree in s =

*r

i=1s_i satisfying a relation of the form b_f(s)

(r i=1

f_i^si =

"r k=1

P_kt_k (r i=1

f_i^si, (2.1)

where P1, . . . , P_r ∈ D[s^ij | i, j ∈ {1, . . . , r}].

In [BMS06a,Section 2], Budur, Musta¸t˘a and Saito show that:

1. bf(s) is a nonzero polynomial,

2. bf(s) is independent of the generating set of I .

2.1. An Alternative Way to Define the Bernstein–Sato Polynomial

In [BMS06a, Section 2.10], Budur, Musta¸t˘a and Saito give the following way to compute the b-function.

For c = (c¹, . . . , c_r) ∈ Z^r, define nsupp(c) = {i | cⁱ < 0}. The Bernstein–

Sato polynomial bf(s) is the monic polynomial of the smallest degree such that b_f(s)+_r

i=1f_i^si belongs to the D[s¹, . . . , s_r]-submodule generated by (

i∈nsupp(c)

, s_i

−cⁱ -

· (r i=1

f_i^si+ci, (2.2)

where c= (c¹, . . . , c_r)runs over the elements ofZ^r such that *_r

i=1c_i = 1. Here s=*r

i=1s_i and._si

m

/= si(s_i− 1) · · · (si− m + 1)/m!.

The advantage of this approach is that we now work in the ring D[s¹, . . . , s_r] and need no longer consider the operators sij or tk.

2.2. A Monomial-Ideal-Specific Way to Define the Bernstein–Sato Polynomial The definition given in the previous subsection can be further specialized to the case of monomial ideals.

Let a be a monomial ideal in k[x¹, . . . , x_n] whose minimal monomial genera- tors are x^α1, . . . , x^αr. Set ℓ_α(s1, . . . , s_r)= s¹α1+ · · · + srα_r, and let ℓ_i(s)be the ith coordinate of this vector. For v∈ Z^r, let psupp(v)= {i | vⁱ >0}.

For c= (c¹, . . . , c_r)∈ Z^r such that*_r

i=1c_i = 1, set g_c(s₁, . . . , s_r)= (

j∈nsupp(c)

, s_j

−c^j -

· (

i∈psupp(ℓα(c))

,ℓ_i(s₁, . . . , s_r)+ ℓi(c) ℓ_i(c)

- , and define I_a to be the ideal ofQ[s¹, . . . , s_r] generated by the polynomials g^c.

Then [BMS06a,Prop. 4.2] asserts that bf(s)is the monic polynomial of small- est degree such that bf(*r

i=1s_i)belongs to the ideal I_a.

3. Bernstein–Sato Polynomials on Toric Varieties

The goal of this section is to extend Definition 2.1 to the toric setting. We first recall some background 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 a commutative ring k, Grothendieck’s ring of k-linear differential operators Dk(R) of R [Gro67] is the R-subalgebra of Hom_k(R, R)defined by

D_k(R)= 0

i∈N∪{0}

D_i,

where D0 = R and Dⁱ = {f ∈ Hom^k(R, R)| f r − rf ∈ Dⁱ−1 for all r ∈ R} for i ≥ 1. When R is the polynomial ring in n variables over an algebraically closed field k of characteristic 0, the ring Dk(R) coincides with the nth Weyl algebra.

Moreover, it is well known that if R is regular over k, then D_k(R) is the R- algebra generated by the k-derivations of R. The most influential applications of D-module theory and, in particular, the b-functions of [BMS06a] are built in this setting.

When R is not regular, the ring Dk(R) is not well-behaved [BGG72].

Nonetheless, when R = k[NA] is a toric algebra over an algebraically closed field k of characteristic 0, there exists an explicit combinatorial description of D_k(k[NA]). This expression of D_k(k[NA]) was obtained independently in [Mus87] and [Jon94] when NA is normal and was further extended to the not necessarily normal case in [ST01]. We denote the ring of differential operators of the toric algebra k[NA] by

D_A= D_k(k[NA]).

One way to understand DA is to identify k[NA] as a subring of the Laurent polynomial ring k[y₁^±1, . . . , y_d^±1] and study the k-linear differential operators of

k[y₁^±1, . . . , y_d^±1]. It is known that if S is a multiplicatively closed subset of a k- algebra R, then Dk(S⁻¹R) ∼= S⁻¹R⊗^RD_k(R). It follows that

D_k(k[y₁^±1, . . . , y_d^±1]) = k[y₁^±1, . . . , y_d^±1]⟨∂^y1, . . . , ∂_yd⟩ is a localization of the nth Weyl algebra. Moreover, one can realize

D_A= {δ ∈ Dk(k[y₁^±1, . . . , y_d^±1]) | δ(k[NA]) ⊆ k[NA]}

as a subring of D_k(k[y₁^±1, . . . , y_d^±1]). In particular, we have the Z^d-graded ex- pression

D_A= !

u∈Z^d

y^u{f (θ) ∈ k[θ¹, . . . , θ_d] | f vanishes on NA \ (−u + NA)}, (3.1) where θ1, . . . , θ_d are the commuting operators yi∂_yi for i= 1, . . . , d. Since we are only concerned with the case whereNA is normal, for a given u ∈ Z^d, the ideal

{f (θ) ∈ k[θ¹, . . . , θ_d] | f vanishes on NA \ (−u + NA)}

is the principal ideal inC[θ1, . . . , θ_d] generated by (

Fσ(u)>0

F_σ((u)−1 j=0

(F_σ(θ₁, . . . , θ_d)− j), (3.2) where the product runs through all facets σ (with F_σ(u) >0) of the cone C asso- ciated with NA, and Fσ is the primitive integral support function of σ as defined in Notation1.2.1.

We are now ready to define the b-function for an ideal I ink[NA]. To make the exposition more transparent, we first treat the case where I is a monomial ideal.

3.2. Monomial Ideals in Semigroup Rings

Let a be a monomial ideal in k[NA] whose minimal monomial generators are Laurent monomials y^β1, . . . , y^βr. For f = (y^β1, . . . , y^βr), we define the Bernstein–Sato polynomial bf(s) exactly as in Definition 2.1, except that the Weyl algebra D is replaced by the ring of differential operators DA on k[NA].

If no polynomial satisfying the definition exists, then we make the convention that bf(s)is zero; however, we will prove later that this is never the case.

In Section2.1, we recalled how in [BMS06a] the computation of b-functions is reduced to a computation over D[s¹, . . . , s_r] (rather than over D[sij]). The same argument can be applied to the toric setting. This implies that Bernstein-Sato poly- nomials over normal toric varieties can be computed using only operators from D_A[s¹, . . . , s_r], without needing to work over the larger ring DA[sij].

Our task becomes to translate the situation from Section2.2to the new setting with monomial ideals in semigroup rings, rather than with monomial ideals in polynomial rings.

Let c∈ Z^r whose coordinates sum to 1, and for β = (β¹, . . . , β_r)∈ (Z^d)^r, set ℓ_β(s₁, . . . , s_r)= s¹β₁+ · · · + s^rβ_r.

To find the Bernstein–Sato polynomial, we need to apply an operator in D_A[s¹, . . . , s_r] to +

i∈nsupp(c)

. _s1

−ci

/·+_r

i=1(y^βi)^si+ci in such a way that the out- come is a multiple of y^*ri=1siβi = y^{ℓβ(s1,...,sr)}.

If we apply the operator from (3.2) with u = ℓβ(c) to +

i∈nsupp(c)

. _si

−ci

/ · +_r

i=1(y^βi)^si+ci, then we obtain (

i∈nsupp(c)

, s_i

−cⁱ

- (

Fσ(ℓβ(c))>0

Fσ(ℓ(β(c))−1 j=0

(F_σ(ℓ_β(s1, . . . , s_r)+ ℓβ(c))− j)

· y^{ℓβ(s1,...,sr)}.

Proposition 3.1. We use the notation introduced before. The Bernstein–Sato polynomial b_f(s) for f = (y^β1, . . . , y^βr) considered as a monomial ideal in k[NA] is the monic polynomial of smallest degree such that bf(s₁ + · · · + s^r) belongs to the ideal generated by

(

i∈nsupp(c)

, s_i

−ci

- (

Fσ(ℓ_β(c))>0

,F_σ(ℓ_β(s₁, . . . , s_r)+ ℓ^β(c)) F_σ(ℓ_β(c))

-

, (3.3)

where c∈ Z^r has the coordinate sum1.

Proof. It remains to be shown that, for any other operator (in DA[s¹, . . . , s_r]) that applied to +

i∈nsupp(c)

._s1

−ci

/·+r

i=1(y^βi)^si+ci yields a multiple of y^{ℓ(s1,...,sr)}, this multiple belongs to the ideal generated by (3.3).

This follows from the description of DA given by (3.1) and (3.2). ! Recall that F is the collection of facets of the cone C of NA 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 inclusionk[NA] → k[N^F].

Proposition 3.2. Denote by F (f ) the sequence of monomials (x^{F (β1)}, . . . , x^{F (βr)}) ink[N^F]. The Bernstein–Sato polynomial b^{F (f )}(s) of F (f ) in the poly- nomial ring k[N^F] coincides with the Bernstein–Sato polynomial bf(s) of f = (x^β1, . . . , x^βr) in k[NA]. In particular, the latter polynomial bf(s) is nonzero, and its roots can be computed using the combinatorial description [BMS06b, Thm. 1.1] applied to⟨x^{F (β1)}, . . . , x^{F (βr)}⟩ ⊆ k[N^F].

Proof. First, we observe that the minimal monomial generators of ⟨x^{F (β1)}, . . . , x^{F (βr)}⟩ ⊆ k[N^F] are the monomials x^{F (β1)}, . . . , x^{F (βr)}. To see this, note that x^{F (βi)} divides x^{F (βj)} is equivalent to Fσ(β_i) ≤ F^σ(β_j) for all facets σ of the cone C. This implies that β_j − βi lies in the cone C. Since β_j − βi∈ Z^r andNA is normal, we see that βj − βⁱ∈ NA, and therefore y^βi divides y^βj ink[NA].

Let αi = F (βⁱ) for i = 1, . . . , r, let ℓ^α(s₁, . . . , s_r) = s¹α₁ + · · · + s^rα_r, and denote by ℓi(s₁, . . . , s_r) the ith coordinate of the vector ℓα(s₁, . . . , s_r).

By [BMS06a, Prop. 4.2] the Bernstein–Sato polynomial of ⟨x^{F (β1)}, . . . , x^{F (βr)}⟩ is the monic polynomial p of smallest degree such that p(s1+ · · · + s^r)lies in the

ideal generated by (

j∈nsupp(c)

, s_j

−c^j

- (

i∈psupp(ℓα(c))

,ℓ_i(s1, . . . , s_r)+ ℓi(c) ℓ_i(c)

-

(3.4) for c∈ Z^r with coordinate sum 1. But note that by construction ℓα = F ◦ ℓ^β, so that the generators (3.4) coincide exactly with the generators (3.3). As a reality check, note that both sets of generators belong to the ring k[s1, . . . , s_r], which does not depend on the ambient rings of the monomial ideals involved. !

3.3. The General Case

Again, recall that we have a linear map F : R^d → R^F with F (NA) ⊆ N^F. We construct a ring homomorphism

k[NA] → k[N^F] via

"k j=1

λ_jy^βj *→

"k j=1

λ_jx^{F (βj)}, (3.5) where β1, . . . , β_k ∈ NA. That this is a homomorphism follows from the 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[NA] is an ideal, then its image F (I) is not an ideal in k[N^F]; we denote by⟨F (I)⟩ the ideal in k[N^F] generated by F (I).

Lemma 3.3. Let I ⊂ k[NA] be the ideal generated by g¹, . . . , g_r ∈ k[NA]. Then F (g₁), . . . , F (g_r)generate⟨F (I)⟩.

Proof. Since g1, . . . , g_r generate I , and F is a ring homomorphism, then any element of F (I ) is obtained as a combination of F (g1), . . . , F (g_r) with coeffi- cients in k[N^F]. This implies that the polynomials F (g¹), . . . , F (g_r) generate

⟨F (I)⟩. !

We apply Definition2.1to a sequence of polynomials in k[NA] using the ring of differential operators DA instead of D. The following is the main result of this section.

Theorem 3.4. Let I ⊂ k[NA] be an ideal, and let g¹, . . . , g_r be generators for I. The Bernstein–Sato polynomial of g = (g¹, . . . , g_r) in k[NA] equals the Bernstein–Sato polynomial of f = (f¹, . . . , f_r)= F (g) = (F (g¹), . . . , F (g_r)) in k[N^F]. Consequently, this Bernstein–Sato polynomial is nonzero and depends only on I, not on the particular set of generators chosen.

Proof. We know that the Bernstein–Sato polynomial bg(s) is the monic polyno- mial of smallest degree such that, for s= s¹+ · · · + s^r, bg(s)+r

i=1g_i^si belongs to the DA[s¹, . . . , s_r]-submodule generated by

(

i∈nsupp(c)

, s_i

−cⁱ -

· (r i=1

g_i^si+ci for (c1, . . . , c_r)∈ Z^r with

"r i=1

c_i = 1,

whereas bf(s) is is the monic polynomial of smallest degree such that b_f(s)+r

i=1f_i^si belongs to the D[s¹, . . . , s_r]-submodule generated by (

i∈nsupp(c)

, s_i

−ci

-

· (r i=1

f_i^si+ci for (c1, . . . , c_r)∈ Z^r with

"r i=1

c_i= 1.

Here D is the ring of differential operators on k[N^F], and DA is the ring of differential operators onk[NA].

Fix c ∈ Z^r such that *

c_i = 1, and let P ∈ D[s¹, . . . , s_r] be such that P[+

i∈nsupp(c)

. _si

−ci

/·+_r

i=1f_i^si+ci] is a polynomial in s¹, . . . , s_r times +_r

i=1f_i^si. Applying the description given by (3.1) and (3.2) to D (instead of DA), we see that we can write P as a finite sum

P = "

u∈Z^F

q_u(s₁, . . . , s_r)x^u' (

uσ<0

−u(^σ−1 j=0

((θ_x)_σ − j) )

p_u(θ_x), where the pu are polynomials in|F | indeterminates with coefficients in k.

Note that, by construction, the element +

i∈nsupp(c)

. _si

−ci

/ · +r

i=1f_i^si+ci is F (NA)-graded. Since P [+

i∈nsupp(c)

. _si

−ci

/ · +_r

i=1f_i^si+ci] is a multiple of +r

i=1f_i^si, we may assume that the operator P is F (NA)-graded as well. In other words, we may assume that u∈ F (Z^d)if qup_u̸= 0.

Thus we rewrite P = "

u=F (v)∈F (Z^d)

q_u(s₁, . . . , s_r)x^u' (

u_σ<0

−u(^σ−1 j=0

((θ_x)_σ − j) )

p_u(θ_x),

and if we denote Pˆ = "

v∈Z^d

q_{F (v)}(s1, . . . , s_r)y^v' (

Fσ(v)<0

−Fσ((v)−1 j=0

(F_σ(θ_y)−j) )

p_{F (v)}((F_σ(θ_y))_σ∈F), then ˆP is an element of D_A[s¹, . . . , s_r], and ˆP applied to +

i∈nsupp(c)

. _si

−ci

/ · +r

i=1g_i^si+ci is a polynomial of s1, . . . , s_r times +r

i=1g^s_iⁱ, and this is the same polynomial that we obtain when we apply P to +

i∈nsupp(c)

. _si

−ci

/· +_r

i=1f_i^si+ci (and divide by+

f_i^si).

Conversely, if we apply an element of DA[s¹, . . . , s_r] to +

i∈nsupp(c)

. _si

−ci

/· +_r

i=1g_i^si+ci and obtain a polynomial in s1, . . . , s_r times +_r

i=1g_i^si, then we can obtain an element of D[s¹, . . . , s_r] that applied to+

i∈nsupp(c)

._si

−ci

/·+r

i=1f_i^si+ci gives exactly the same polynomial in s1, . . . , s_r times +r

i=1f_i^si. To do this, we write the element of DA[s¹, . . . , s_r] as a sum of terms of the form

q_v(s₁, . . . , s_r)y^v' (

Fσ(v)<0

−F^σ((v)−1 j=0

(F_σ(θ_y)− j) )

p_v(θ_y),

then express (nonuniquely) the polynomial pv as sums of powers of the linear forms Fσ(θ_y), and then apply F in the obvious way. Nonuniqueness comes be- cause we need to choose d linearly independent forms F_σ in order to get k- algebra generators of k[θy]; however, since the image of F as a linear map is d-dimensional, the choice does not affect the image. ! Example 3.5. In [BMS06a], their b-function is defined for any scheme Z. Up to a shift by codimension, it is independent of the choice of the embedding of Z into a smooth variety. We provide a simple example showing that their b-functions are quite different from ours. This was suggested to us by Nero Budur.

Consider the affine line Z = {u = v = 0} embedded in the toric hypersurface X= {uw − v² = 0}. The b-function of the scheme Z in the sense of Budur, Mus- ta¸t˘a and Saito is bZ(s)= s. However, using our definition and Theorem 3.4, we see that the toric b-function of Z embedded in X is the same as the Budur–

Musta¸t˘a–Saito b-function of the subscheme {x² = xy = 0} of the affine plane, which is equal to (s+ 1)²(s+ 3/2).

4. Multiplier Ideals on Normal Toric Varieties

We recall basic definitions of multiplier ideals that can be found in Lazarsfeld’s text [Laz04].

Let X be a smooth variety over an algebraically closed fieldk of characteristic 0. Let I ⊆ OX be an ideal sheaf, and let α > 0 be a rational number. Fix a log resolution µ: X^′→ X of I with I · OX^′ = OX^′(−E). The multiplier ideal of the pair (X, αI) is defined as

J (X, αI) = µ∗OX^′(K_X′/X− ⌊α · E⌋),

where K_X′/X = KX^′ − µ^∗K_X is the relative canonical divisor of µ, which is an effective divisor supported on the exceptional locus of µ whose local equation is given by the determinant of the derivative dµ. The definition does not depend on the choice of log resolution. One of the important features of multiplier ideals is that J (X, αI) measures the singularity of the pair (X, αI): a smaller multiplier ideal corresponds to a worse singularity.

Notice that J (X, αI) becomes smaller as α increases. The jumping coeffi- cients of (X, αI) are the positive real numbers 0 < α¹ < α2 < · · · such that J (X, α^jI) = J (X, αI) ̸= J (X, α^j+1I) for α^j ≤ α < α^j+1 (j ≥ 0) where α₀ = 0. When X is affine and I is the ideal in the coordinate ring k[X] corre- sponding the the sheaf I, Budur, Musta¸t˘a and Saito proved that the jumping coef- ficients of (X, αI ) in[α^f, α_f + 1) are roots of b^f(−s), where f = (f¹, . . . , f_r)is a set of generators for I , bf(s) is the Bernstein–Sato polynomial of I , and αf is the smallest root of bf(−s). One of our goals is to generalize this correspondence between b-function roots and jumping coefficients to monomial ideals on affine normal toric varieties.

In the particular case that I is a monomial ideal in the polynomial ring k[x¹, . . . , x_m], Howald [How01] gave the following combinatorial formula for

the multiplier ideals of the pair (A^m, αI ):

J (A^m, αI )= ⟨x^v | v + e ∈ int(αP^I)⟩,

where P_I is the Newton polyhedron of I , and e is the vector (1, 1, . . . , 1) inN^m. In particular, a rational number α > 0 is a jumping coefficient of (A^m, αI ) if and only if the boundary of−e + int(αP^I)contains a lattice point inN^m.

The notion of multiplier ideal can be generalized to the case that I is an ideal sheaf on a normal variety 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 let α > 0 be a rational number. Suppose that I · OX^′ = OX^′(−E). Then we can define the multiplier ideal J (X, +, αI) associated with the triple (X, +, αI) as

J (X, +, αI) = µ∗O^X′(K_X′ − ⌊µ^∗(K_X + +) + αE⌋).

Again, this definition is not dependent on the choice of µ. De Fernex and Ha- con [dFH09] have given a definition of J (X, αZ) for non-Q-Gorenstein X with- out using the boundary divisor +. They showed that there exists a boundary divi- sor + 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 element of the set

{J (X, +, αZ) | + is a boundary divisor}.

4.1. A New Expression for Multiplier Ideals on Toric Varieties

Let us explain how we use the map F to understand the multiplier ideals of de Fernex and Hacon in the case of monomial ideals on normal toric varieties.

Letk[NA] ⊂ k[y₁^±, . . . , y_d^±] be a normal semigroup ring in our setting, and let X= Spec(k[NA]). Each facet σ ∈ F corresponds to a torus invariant prime Weil divisor Dσ on X. The canonical class of X is represented by the torus invariant canonical divisor KX = −*

σ∈F D_σ. A boundary divisor + is an effective Q- divisor such that KX+ + is Q-Cartier, which means there exist l ∈ Z and u ∈ Z^d such that l(KX+ +) = div(y^u). Denote w+= ^u_l. Notice that

+= "

σ∈F

(1+ F^σ(w₊))D_σ,

so the effectivity of + is equivalent to the condition that F_σ(w₊)≥ −1 for all σ ∈ F . Conversely, each w ∈ Q^d such that Fσ(w)≥ −1 for all σ ∈ F gives rise to a boundary divisor +w on X by the same formula

+_w = "

σ∈F

(1+ Fσ(w))D_σ.

Let I be a monomial ideal in k[NA], and let α > 0 be a rational number.

Blickle [Bli04] showed that the multiplier ideal J (X, +, αI) is equal to the ideal

⟨y^v ∈ k[NA] | v − w⁺ ∈ int(αP^I)⟩, generalizing Howald’s description of mul- tiplier ideals [How01] in the case of monomial ideals in polynomial rings. Be

cautioned about the mistake of the sign of w+ in Blickle’s original description.

For w∈ Q^d, denote

,_w,αI = [w + int(αP^I)] and ,^αI = 0

w∈Q^d:F^σ(w)≥−1 ∀σ ∈F

,_w,αI. (4.1) Then the multiplier ideal of de Fernex and Hacon is

J (X, αI ) = ⟨y^v ∈ k[NA] | v ∈ ,^αI⟩. (4.2) We claim that J (X, αI) can be computed using an analog of Howald’s for- mula. Recall that the linear map F : R^d → R^F induces ring homomorphisms

k[NA]−→ k[F (NA)] → k[N^{∼ F}].

By abusing notation denote

F (I )= k[F (NA)] · ⟨x^{F (v)}| y^v ∈ I⟩

the ideal in k[F (NA)] obtained from the semigroup isomorphism F : NA−→^∼ F (NA). The monomial ideal in k[N^F] generated by F (I) is denoted by

J = k[N^F] · ⟨F (I)⟩.

The Newton polyhedron of F (I ) is denoted by PF (I ); this is the convex hull of{F (v) | y^v ∈ I} in F (R^d). We have

P_{F (I )}= conv' 0

v:y^v∈I

(F (v)+ F (NA)) )

. On the other hand, the Newton polyhedron of J inR^F is

P_J = conv' 0

v:y^v∈I

(F (v)+ N^F) )

.

Note that if y^β1, . . . , y^βr is the set of minimal monomial generators of I , then {β¹, . . . , β_r} (respectively, {F (β¹), . . . , F (β_r)}) is exactly the set of vertices of the Newton polyhedron PI (respectively, PF (I )or PJ). We have the following lemma that compares the relative interiors of αP_{F (I )} and αP_J.

Lemma 4.1. For generalNA, the relative interiors of P^{F (I )} and P_J satisfy int(αPF (I ))= 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 face of PF (I ). Similarly, a point u lies in int(PJ) if and only if u− v ∈ int R^F_≥0 for some v on a bounded face of PJ. Since F (I ) and J have the same minimal monomial generators, the bounded faces of P_{F (I )}and PJ coincide.

Therefore, it suffices to show that

int(F (C))= F (C) ∩ int(R^F_≥0).

But for p ∈ C, F (p) ∈ int F (C) if and only if p is not contained in any facet

σ ∈ F , which means exactly F^σ(p) >0. !

Let e be the element inR^F such that eσ = 1 for all σ ∈ F .

Proposition 4.2. The multiplier ideal J (X, αI) is equal to the ideal

⟨y^v ∈ k[NA] | F (v) + e ∈ int(αP^J)⟩.

Proof. By (4.1) and (4.2) it suffices to show that

F (,_αI ∩ NA) = (−e + int(αP^J))∩ F (NA).

For the containment F (,αI ∩ NA) ⊆ (−e + int(αP^J))∩ F (NA), it is enough to verify that

F (w+ int(αP^I))⊆ [−e + int(αP^J)]

for any w ∈ Q^d satisfying Fσ(w) ≥ −1 for all σ ∈ F . Any such w satisfies F (w)+ e ∈ R^F_≥0, so by Lemma4.1

F (w+ int(αPI))= [F (w) + int(αF (PI))]

= [F (w) + int(αPF (I ))]

⊆ [F (w) + int(αPJ)]

= [−e + (F (w) + e) + int(αP^J)]

⊆ [−e + R^F_≥0+ int(αP^J)]

⊆ [−e + int(αP^J)].

For the other containment F (,_αI ∩ NA) ⊇ (−e + int(αPJ)) ∩ F (NA), let v∈ NA be such that F (v) + e ∈ int(αP^J). Then there exists u lying on a bounded face of αPI such that

F (v)+ e − F (u) ∈ int(R^F_≥0).

In particular, F_σ(v− u) > −1 for all σ ∈ F . Take any p ∈ int C and ϵ > 0 small enough so that

w:= v − u − ϵp ∈ Q^d and F_σ(v− u − ϵp) > −1.

Then F (v)− F (u) − F (w) = F (ϵp) ∈ int C where F (u) lies on a bounded face of αPF (I ). Therefore, we have F (v)− F (w) ∈ int(αP^{F (I )}), and hence F (v)∈

F (,_w,αI ∩ NA) ⊆ F (,^αI ∩ NA), as desired. !

Remark 4.3. Notice that e may not be in F (NA) and that e ∈ F (NA) if and only if X is Gorenstein. Even if we extend to rational coefficients, then the element e may not be in F (Q ⊗ NA) = F (Q^d). The condition e∈ F (Q^d) holds exactly when X isQ-Gorenstein, namely K^X is aQ-Cartier divisor. This is the case where XisQ-factorial (or, equivalently, the semigroup NA is simplicial).

The new expression of multiplier ideals in Proposition 4.2 is motivated by an observation in the special case where X is Q-Gorenstein. In that case, we have e ∈ F (Q^d) and ,αI = ,F⁻¹(−e),αI. So w = F⁻¹(−e) corresponds to the + of de Fernex and Hacon. In fact, the divisor +_F−1(−e)= 0 coincides with the

canonical choice of boundary divisor for the pair (X, αI ) in the Q-Gorenstein case. Therefore, by (4.2) the multiplier ideal

J (X, αI ) = ⟨y^v ∈ k[NA] | v ∈ ,^αI⟩

= ⟨y^v ∈ k[NA] | v ∈ ,F⁻¹(−e),αI⟩

= ⟨y^v ∈ k[NA] | F (v) + e ∈ int(αF (P^I))⟩

= ⟨y^v ∈ k[NA] | F (v) + e ∈ int(αP^{F (I )})⟩.

4.2. Roots–Jumping Coefficients Correspondence on Toric Varieties Now, we combine the previous observations to establish the following theorem.

Theorem 4.4. Let NA be a normal semigroup, let X = Spec(k[NA]) be its as- sociated affine toric variety, and let I be a monomial ideal on X. Suppose αI is the smallest root of b_I(−s) where b^I(s)is the Bernstein–Sato polynomial of I in k[NA]. Then all jumping coefficients of the pair (X, I) in [αI, α_I + 1) are roots of b_I(−s). Moreover, the number αI is the smallest jumping coefficient (i.e. the log-canonical threshold) of (X, I ).

Proof. By Proposition 3.2 the Bernstein–Sato polynomial bI(s) of I coincides with the Bernstein–Sato polynomial of the monomial ideal J = k[N^F] · ⟨F (I)⟩.

Thus by [BMS06a, Thm. 2], the jumping 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 jump- ing coefficient of (A^F, J )exactly when the boundary of (−e + αP^J)contains a lattice point in N^F. Also, according to Proposition4.2, a number α is a jumping coefficient of (X, I ) exactly when the boundary of (−e + αP^J) contains a lat- tice point in F (NA). Therefore, the jumping coefficients of (X, I) are jumping coefficients of (A^F, J ), and the first statement of this theorem follows.

To prove αI is the log-canonical threshold of (X, I ), it suffices to show that the boundary of (−e + α^IP_J) intersects F (NA). Since αI is the log-canonical threshold of (A^F, J ), we have

1∈ k[N^F] = J (A^F, α_IJ ), and hence F (0)= 0 ∈ (−e + α^IP_J)∩ F (NA).

! Example 4.5. Let A = (^{1 1 1 1}_{0 1 2 3}) and I = ⟨y¹y₂, y₁y₂²⟩ a monomial ideal in k[NA]. In this case, the semigroup ring k[NA] is simplicial and hence Q- Gorenstein. The linear mapping F : R²→ R² is represented by the matrix (³₀ ⁻¹₁).

The subsemigroup k[F (NA)] of k[x¹, x₂] is generated by x₁³, x₁²x₂, x1x₂², and x₂³ and by the monomial ideal J = k[x¹, x₂]⟨x1²x₂, x₁x₂²⟩. Using the method of Budur, Musta¸t˘a and Saito as discussed in Section 2.2, we can compute the Bernstein–Sato polynomial b_J(s)= (s + 1)²(3s+ 2)(3s + 4). (We point out that b-function algorithms have been developed and implemented; see [BL10].) More- over, by Proposition3.2we have bI(s)= (s + 1)²(3s+ 2)(3s + 4) as well. On the

other hand, using Howald’s formula, we find that ²₃,1, and ⁴₃ are jumping coeffi- cients of (A², J ), but only ²₃ and 1 are jumping coefficients of (X, I ) according to Proposition4.2.

Acknowledgments. We are grateful to the referees for their valuable com- ments and suggestions. We also thank Nero Budur, whose observations on the [BMS06a] definition of Bernstein–Sato polynomials for arbitrary schemes led us to Example3.5. The first author thanks Texas A&M University for the hospi- tality he enjoyed during his visit in January 2016 when this work was initiated.

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J.-C. Hsiao

Mathematics Department

National Cheng Kung University Tainan City 70101

Taiwan

jhsiao@mail.ncku.edu.tw

L. F. Matusevich

Mathematics Department Texas A&M University College Station, TX 77843 USA

laura@math.tamu.edu