aa r X i v : . [ m a t h . AG ] F e b AN ALGORITHM FOR HODGE IDEALS
GUILLEM BLANCO
Abstract.
We present an algorithm to compute the Hodge ideals [MP19a, MP19b] of Q -divisors associated to any reduced effective divisor D . The computation of the Hodgeideals is based on an algorithm to compute parts of the V -filtration of Malgrange andKashiwara on ι + O X ( ∗ D ) and the characterization [MP20b] of the Hodge ideals in termsof this V -filtration. In particular, this gives a new algorithm to compute the multiplierideals and the jumping numbers of any effective divisor. Introduction
Let X be a smooth complex variety and let D be a reduced effective divisor on X .Consider O X ( ∗ D ) the sheaf of meromorphic functions with poles along the divisor D .This is a coherent left D X -module that underlies a mixed Hodge module j ∗ Q HU [ n ], where U := X \ D , j : U ֒ → X is the open inclusion, and Q HU [ n ] is the constant pure Hodgemodule on U , see [Sai88, Sai90]. Any D X -module associated to a mixed Hodge modulecarries a good filtration F • , the Hodge filtration of the mixed Hodge module.It is shown in [Sai93] that the Hodge filtration on O X ( ∗ D ) is contained in the poleorder filtration, namely F k O X ( ∗ D ) ⊆ O X (( k + 1) D ) for all k ≥ . In order to study the Hodge filtration on O X ( ∗ D ), Mustat¸ˇa and Popa [MP19a] introduceda set of ideal sheaves, the Hodge ideals I k ( D ) of D , that are defined by F k O X ( ∗ D ) = I k ( D ) ⊗ O X O X (( k + 1) D ) , for all k ≥ . In a subsequent work, Mustat¸ˇa and Popa [MP19b] generalized the notion of Hodge idealsto arbitrary Q -divisors. If f ∈ O X ( X ) is a global regular function, denote D := div( f )and let Z be the support of D . Then, for α ∈ Q > one can associate to this data a twistedversion of the localization D X -module considered above, namely M ( f − α ) := O X ( ∗ Z ) f − α , that is, a rank one free O X ( ∗ Z )-module with generator the formal symbol f − α and wherethe action of a derivation is the usual one. This D X -module can be endowed with afiltration F k M ( f − α ) , k ≥ D -moduleunderlying a mixed Hodge module on X , see [MP19b, § Q -divisor αD are defined by F k M ( f − α ) = I k ( D ) ⊗ O X O X ( kZ ) f − α , for all k ≥ . Determining the Hodge ideals of a given divisor is a notoriously difficult problem. Inthe case that the divisor D has quasi-homogeneous isolated singularities, the Hodge idealsof Q -divisors have been computed by Zhang [Zha18]. For the case of a free divisor D , The author is supported by a Postdoctoral Fellowship of the Research Foundation – Flanders. the Hodge ideals I k ( D ) are determined by Casta˜no, Narv´aez Macarro and Sevenheck[CMS19]. The first Hodge ideal of the Q -divisor αD coincides with the multiplier ideal J (( α − ε ) D ) , < ε ≪
1, see [MP19b, Proposition 9.1]. Hence, the Hodge ideals can beseen as a generalization of multiplier ideals. For the case of multiplier ideals, there arealgorithms to compute them due to Berkesch and Leykin [BL10] and Shibuta [Shi11].In this work, we will use the characterization given in [MP20b] of the Hodge ideals I k ( αD ) associated to a reduced effective divisor D in terms of the V -filtration of Malgrangeand Kashiwara [Mal83, Kas83]. Given the V -filtration V α ι + O X , α ∈ Q ∩ (0 ,
1] on the D -module theoretic direct image ι + O X of O X by the graph embedding ι of f , Algorithm 1will compute a set of generators for the O X -module G p V α ι + O X = V α ι + O X ∩ p M j =0 O X ∂ jt δ = (cid:26) p X j =0 v j ∂ jt δ ∈ V α ι + O X (cid:12)(cid:12)(cid:12) v j ∈ O X (cid:27) for a fixed p ∈ N using Gr¨obner basis techniques in the Weyl algebra. After [MP20b,Theorem A ′ ], this determines generators for the Hodge ideals I k ( αD ) , k = 0 , . . . , p . More-over, by a result of Budur and Saito [BS05], Algorithm 1 gives also a new procedure tocompute the multiplier ideals and the (global) jumping numbers of any effective divisor,not necessarily reduced.This work is organized as follows. In Section 2 we review the results related to thetheory of V -filtrations of Malgrange and Kashiwara and the Bernstein-Sato polynomialsthat will be needed for the main algorithm. In Section 3, we present Algorithm 1 and weprove its correctness. Some non-trivial examples of Hodge ideals are included at the end.The algorithms described in this work have been implemented in the computer algebrasystem Singular [DGPS21].2.
The V -filtration of Malgrange and Kashiwara Let X be a smooth complex variety of dimension n with structure sheaf O X . Let D X denote the sheaf of differential operators on X . The V -filtration of Malgrange [Mal83] andKashiwara [Kas83] on a D X -module is defined with respect to a closed subvariety Z ⊂ X .Through this work we will assume that Z is a codimension one subvariety globally definedby a regular function f ∈ Γ( X, O X ).1. The smooth case.
When Z is a smooth subvariety, the V -filtration on D X along Z is defined by V k D X = { P ∈ D X | P · ( f ) i ⊂ ( f ) i + k for all i ∈ Z } with k ∈ Z and ( f ) i = O X for i ≤ X, Z ) is smooth, one can consider local algebraic coordinates of theform x , . . . , x n − , t = f . Therefore, V k D X is locally generated over O X by Y ≤ i ≤ n − ∂ α i x i t ν ∂ µt , with ν − µ ≥ k. From this, the V -filtration on D X along Z is then an exhaustive decreasing filtrationsatisfying V i D X · V j D X ⊆ V i + j D X with equality for i, j ≥
0. In the sequel, t will alwaysdenote a local equation for Z and ∂ t a local vector field such that [ ∂ t , t ] = 1. N ALGORITHM FOR HODGE IDEALS 3
In general, a V -filtration on a coherent left D X -module M along Z ⊂ X is a rationalfiltration ( V α M ) α ∈ Q that is exhaustive, decreasing, discrete and left continuous, such thatfollowing conditions are satisfied:i) Each V α M is a coherent module over V D X .ii) For every α ∈ Q , there is an inclusion t · V α M ⊆ V α +1 M with equality for α > α ∈ Q , one has ∂ t · V α M ⊆ V α − M . In particular, V i D X · V α M ⊆ V α + i M .iv) For every α ∈ Q , the action of ∂ t t − α is nilpotent onGr αV := V α M /V >α M where V >α M := S α ′ >α V α ′ M .All conditions are independent of the choice of the local coordinate t . In case such afiltration exists then it is necessarily unique, see [Sai88, Lemme 3.1.2]. Under reasonableassumptions for the D X -module M , V -filtrations do exist. Theorem 1.1 ([Kas76, Mal83]) . Let M be a regular holonomic D X -module with quasi-unipotent monodromy around Z . Then, M admits a V -filtration along Z . The graph embedding.
In general, when Z is singular, one reduces to the smoothcase using the graph embedding of f . Namely, let ι : X −→ X × C , x ( x, f ( x ))be the closed embedding defined by the graph of f . Define Y := X × C and let t bethe projection on the second factor of Y . This way, ( t = 0) is the smooth hypersurface X × { } in Y . Given now M a D X -module, we can consider the D -module theoreticdirect image by the graph embedding ι + M := M ⊗ C C [ ∂ t ]with the left D X × C -action given as follows. Let m be a section of M ,i) g · ( m ⊗ ∂ jt ) = gm ⊗ ∂ jt , for g a section of O X .ii) t · ( m ⊗ ∂ jt ) = f m ⊗ ∂ jt − jm ⊗ ∂ j − t .iii) ∂ t · ( m ⊗ ∂ jt ) = m ⊗ ∂ j +1 t .iv) D ( m ⊗ ∂ jt ) = D ( m ) ⊗ ∂ jt − D ( f ) m ⊗ ∂ j +1 t , for D a section of Der( O X ),see for instance [RH08, Example 1.3.5].In this situation one can consider the V -filtration on ι + M along X × { } ∼ = Z . Then,define V • M := M ∩ V • ι + M for the V -filtration on M along Z . Notice also that in thiscase V D Y is just D X h t, t∂ t i .For the case of the D X -module O X one has the following alternative description of ι + O X . There is an isomorphism of D Y -modules ι + O X ∼ = O X [ t ] f − t / O X [ t ] . G. BLANCO
Indeed, denoting by δ the class of f − t in O X [ t ] f − t / O X [ t ], any section can be uniquelywritten as X j ≥ h j ∂ jt δ, with h j being sections of O X and only finitely many terms being non-zero. Any suchsection can be identified with P j ≥ h j ⊗ ∂ jt and one can check that the D Y -action coincide.Notice that by definition one has f δ = tδ .Given an arbitrary D X -module M , one recovers the original definition of ι + M via thefollowing isomorphism of D Y -modules ι + M ∼ = M ⊗ O X ι + O X = M j ≥ M ⊗ O X O X ∂ jt δ. This description of ι + M leads to the following increasing and exhaustive filtration of O X -modules on ι + M that will be useful in the sequel, G k ι + M := k M j =0 M ⊗ O X O X ∂ jt δ, with G k ι + M /G k +1 ι + M ∼ = M as O X -modules.3. Hodge ideals and the V -filtration. The Hodge ideals of a reduced divisor D =div( f ) can be described it terms of the V -filtration on ι + O X . Before presenting the mainresult from [MP20b] that we will use, it is convenient to define the following polynomials, Q i ( X ) = X ( X + 1) · · · ( X + i − ∈ Z [ X ] . Theorem 3.1 ([MP20b, Theorem A ′ ]) . If D is a reduced divisor, then for every positiverational number α , and every p ≥ , one has I p ( αD ) = ( p X j =0 Q j ( α ) f p − j v j (cid:12)(cid:12)(cid:12)(cid:12) p X j =0 v j ∂ jt δ ∈ V α ι + O X ) . If one defines G k V α ι + O X := G k ι + O X ∩ V α ι + O X for all α ∈ Q , then in order to getgenerators for the Hodge ideals I k ( αD ) , k = 0 , . . . , p, it is enough to compute a set ofgenerators for the O X -module G p V α ι + O X .The Hodge ideals I p ( αD ) are a generalization of the multiplier ideals J ( αD ) , α ∈ Q > that usually appear in the context of birational geometry, see [Laz04, § III]. It is a resultof Budur and Saito [BS05] that multiplier ideals have an interpretation in terms of D X -modules via the V -filtration on O X along D . Theorem 3.2 ([BS05, Thm 0.1]) . Let D = div( f ) an effective divisor on X , then J ( αD ) = V α + ǫ O X for 0 < ǫ ≪ . Indeed, I ( αD ) = J (( α − ǫ ) D ) for 0 < ǫ ≪
1, see [MP19b, Proposition 9.1] andTheorem 3.1 can be seen as a generalization of Theorem 3.2 for the case of reduceddivisors.
N ALGORITHM FOR HODGE IDEALS 5 The Bernstein-Sato polynomial.
In order to study the V -filtration on the D Y -module ι + O X it is convenient to work on a bigger D Y -module where multiplication by f is bijective, namely ι + O X ( ∗ D ), where D = div( f ).Under the hypothesis that f acts bijectively on a D X -module M , that is, M hasstructure of O X ( ∗ D )-module, one can show that multiplication by t is bijective in ι + M .Hence, for the particular case of O X ,(4.1) ( ι + O X ) t = ι + O ( ∗ D ) . Denote by M f the natural s = − ∂ t t stable lattice in ι + O X ( ∗ D ), that is M f := D X [ s ] f s ⊆ O X [ f − , s ] f s ∼ = ι + O X ( ∗ D ) , where the symbolic power f s is naturally identified with δ . Notice that, since ∂ t tδ is asection of ι + O X , we have the inclusion M f ⊆ ι + O X .Since, one has the relation ts = ( s + 1) t , multiplication by t leaves invariant M f , thatis t · M f ∼ = D X [ s ] f · f s ⊆ M f . Therefore, M f is in fact a D X h t, s i -module. In addition,it is easy to see that(4.2) ( M f ) t = ( ι + O X ) t . After Equations 4.1 and 4.2, it will be convenient to define the following increasing andexhaustive filtration T k of D X h t, s i -modules on ι + O X ( ∗ D ) by T k ι + O X ( ∗ D ) := t − k M f ∼ = D X [ s ] f s − k . For a general D X -module, one has the following relation that will be needed later on. Proposition 4.1 ([MP20b, Proposition 2.5]) . If M is a D X -module on which f actsbijectively, then we have an isomorphism of D X h t, t − , s i -modules Φ : M [ s ] f s ∼ = −→ ι + M , us j f s u ⊗ ( − ∂ t t ) j δ. The inverse isomorphism Ψ is given by u ⊗ ∂ jt δ uf j Q j ( − s ) f s . This setting leads to one of the fundamental objects in the theory of D -modules. Proposition 4.2 ([Ber72]) . The action of s induces an endomorphism s : M f /t M f −→ M f /t M f , which has a minimal polynomial equal to the Bernstein-Sato polynomial b f ( s ) of f . It is a well-known result due to Kashiwara [Kas76] that the roots of the Bernstein-Satopolynomial are negative rational numbers. The existence of the V -filtration on ι + O X isoriginally due to Malgrange [Mal83] using the rationality of the roots of the Bernstein-Sato polynomial. In fact, Malgrange in [Mal83] proves the existence of the V -filtrationon ι + O X ( ∗ D ). Then, one simply has that V • ι + O X = ι + O X ∩ V • ι + O X ( ∗ D ). Moreover,the following is true. Lemma 4.3 ([Sai88, Lemme 3.1.7]) . The canonical inclusion ι + O X ֒ → ι + O X ( ∗ D ) inducesan equality V α ι + O X = V α ι + O X ( ∗ D ) for all α > . G. BLANCO The algorithm
In this section we shall assume that X = A n C . Therefore, R = Γ( X, O X ) = C [ x , . . . , x n ]is the polynomial ring and D = Γ( X, D X ) = C [ x , . . . , x n ] h ∂ x , . . . , ∂ x n i is the Weyl alge-bra. The algorithm presented in this section will make use of the following constructionsin computational D -module theory.5. The s -parametric annihilator. Let f ∈ R be non-constant. The cyclic D [ s ]-module D [ s ] f s is isomorphic to D [ s ] / Ann D [ s ] f s , where Ann D [ s ] f s is the s -parametric annihilatorof the formal symbol f s .Consider the Malgrange ideal of f , I f := D h t, ∂ t ih f − t, ∂ x + ∂f∂x ∂ t , . . . , ∂ x n + ∂f∂x n ∂ t i . Then, the s -parametric annihilator of f equals I f ∩ D [ ∂ t t ] | ∂ t t = − s , see for instance [SST00,Theorem 5.3.4]. Moreover, such elimination of variables can be computed using Gr¨obnerbasis techniques in the Weyl algebra, see [SST00, Algorithm 5.3.6]. There are similarways to compute generators for Ann D [ s ] f s due to Brian¸con and Maisonobe [BM02] thatusually perform better due to the need to eliminate fewer variables.After Proposition 4.2, the Bernstein-Sato polynomial b f ( s ) of f can then be computedas the minimal polynomial of s acting on D [ s ] f s D [ s ] h f i f s ∼ = D [ s ]Ann D [ s ] f s + D [ s ] h f i . That is, h b f ( s ) i = (Ann D [ s ] f s + D [ s ] h f i ) ∩ C [ s ]. This intersection can either be computedby standard Gr¨obner basis elimination techniques or taking advantage that C [ s ] is aprincipal subalgebra of D [ s ], see [ALMM09].6. Modulo operation.
The following construction from [Lev05] provides an efficientway of computing the kernel of morphisms of D -modules. Let N, M be left submodulesof the free submodules D n = P ni =0 De i and D m , respectively. Consider, φ : D n /N −→ D m /M, e i Φ i , a morphism of left D -modules given by the matrix Φ ∈ D n + m . Define the matrix Y = (cid:18) Φ M Id n × n (cid:19) ∈ D ( n + m ) × ( m + k ) , where k ∈ N is the number of generators of M . Let Z = Y ∩ L n + mi = m +1 De i , this intersectioncan be computed with standard elimination of components. Then, one hasKer φ ∼ = ( Z + N ) /N, see [Lev05, Lemma 9]. This method avoid the computation of unnecessary syzygiesand computes only those relevant to Φ. Keeping the same notation from [Lev05] and[DGPS21], we will denote the operation that computes a system of generators for Z fromgenerators of Φ and M as Modulo (Φ , M ). N ALGORITHM FOR HODGE IDEALS 7 The algorithm.
We present next the main contribution of this work. The followingalgorithm computes a set of generators of the O X -modules G p V α ι + R for any α ∈ Q ∩ (0 , V -filtration, there is only a finitenumber of different such O X -modules when α ranges in Q ∩ (0 , V -filtration on ι + O X ( ∗ D ) by Malgrange [Mal83].The main ideas behind the algorithm are the following. For α ∈ Q ∩ (0 , G p V α ι + R [ f − ]. After (4.2), any element of ι + R [ f − ] ∼ = R [ s, f − ] f s liesin T k ι + R [ f − ] ∼ = t − k M f ∼ = D [ s ] f s − k for some k ∈ N . Then, consider the endomorphism s : D [ s ] f s − p tD [ s ] f s −→ D [ s ] f s − p tD [ s ] f s with minimal polynomial b ( p ) f ( s ). Since the action of t is bijective, the roots of b ( p ) f ( s ) areof the form α + k for α a root of b f ( s ) and k = 0 , . . . , p . Hence, one has the decomposition(7.1) D [ s ] f s − p tD [ s ] f s = M λ P λ , where the sum ranges over all the roots λ of b ( p ) f ( − s ) and P λ is the submodule on which s + λ acts nilpotently. Consider, for α ∈ Q ∩ (0 , D [ s ]-submodule W α satisfying tD [ s ] f s ⊆ W α ⊆ D [ s ] f s − p and(7.2) W α tD [ s ] f s = M λ>α P λ . Then, G p V α R [ f − ] can be compute from W α ∩ R [ s ] by taking all the elements of degreeless than or equal to p in s . The full details of the correctness of the algorithm are delayeduntil Theorem 7.1 below.All the sets appearing in Algorithm 1 below are assumed to be ordered. Algorithm 1. (Generators G p V α ι + R ) Input:
A reduced f ∈ R and p ∈ N . Output:
A basis of the R -module G p V α ι + R for α ∈ Q ∩ (0 , G ← Gr¨obner basis of Ann D [ s ] f s w.r.t. any monomial ordering J ← Gr¨obner basis of D [ s ] h G, f i w.r.t. an elimination order for x, ∂ b f ( s ) ← generator of J ∩ C [ s ] ρ f ← { ( α i , n i ) ∈ Q × N | b f ( s ) = ( s − α ) n · · · ( s − α r ) n r , α < · · · < α r < } J ( p ) ← D [ s ] h G | s s − p , f p +1 i for ( α, n ) ∈ { ( α i + k, n i ) | ( α i , n i ) ∈ ρ f , α i + k < , k = 0 , . . . , p } do K α ← Modulo ( s − α, J ( p ) ) for i = 1 , . . . , n − do K α ← Modulo ( s − α, K α ) end for end for α ′ ← −∞ V α ′ ← S λ< − K λ G. BLANCO for α ∈ { α i + k | ( α i , n i ) ∈ ρ f , α i + k ∈ [ − , , k = 0 , . . . , p } do V α ← Gr¨obner basis of D [ s ] h V α ′ , K α i w.r.t. an elimination order for ∂ H α ← Gr¨obner basis of V α ∩ R [ s ] w.r.t. an elimination order for s H ( p ) α ← (cid:8) P pj =0 h j ( − ∂ t t ) j | P pj =0 s j h j ∈ H α (cid:9) B ( p ) α ← (cid:8) P pj =0 h ′ j ∂ jt f j · f − p | P pj =0 h ′ j ∂ jt t j ∈ H ( p ) α (cid:9) α ′ ← α end for return B ( p ) α Theorem 7.1.
Algorithm 1 is correct.Proof.
In Lines 1–4, the algorithm starts by computing a Gr¨obner basis G of the s -parametric annihilator Ann D [ s ] f s and the Bernstein-Sato polynomial b f ( s ) of f by usingthe standard methods discussed at the beginning of this section.In Line 5, J ( p ) is a basis of the submodule of D [ s ] giving a presentation of D [ s ] f s − p /tD [ s ] f s .Indeed,(7.3) D [ s ] f s − p tD [ s ] f s ∼ = t − p D [ s ] f s tD [ s ] f s = t − p D [ s ] f s t p +1 · t − p D [ s ] f s ∼ = D [ s ]Ann D [ s ] f s − p + D [ s ] h f p +1 i . Notice that, since we already have a basis G of Ann D [ s ] f s , after the substitution s s − p in G one gets a basis of Ann D [ s ] f s − p . Since the action of t is bijective in R [ s, f − ] f s , theminimal polynomial b ( p ) f ( s ) of s acting on (7.3) can be determined from the Bernstein-Satopolynomial b f ( s ) that has been already computed.The loop that starts in Line 6 iterates over the roots of b ( p ) f ( s ) that are strictly negative.For each such a root α having multiplicity n α the next steps, Lines 7–10, compute a basis K α of the kernel of the morphism( s − α ) n α : D [ s ] D [ s ] h J ( p ) i −→ D [ s ] D [ s ] h J ( p ) i . Then, the submodule P − α from (7.1) associated to s − α is isomorphic to D [ s ] h K α i /D [ s ] h J ( p ) i .The computation is done inductively by computing the kernel of ( s − α ) i +1 from the kernelof ( s − α ) i . This strategy makes the computation much more efficient in practice.The last part of the Algorithm works as follows. The sets V α = S λ<α K λ form abasis of the D [ s ]-submodules W − α from (7.2). Indeed, notice that by construction everysubmodule D [ s ] h K α i contains D [ s ] h J ( p ) i and hence it contains D [ s ] h f p +1 i . Then, since tD [ s ] f s ∼ = t p +1 D [ s ] f s − p , the submodules tD [ s ] f s ⊆ W α ⊆ D [ s ] f s − p are isomorphic to D [ s ] h f p +1 i Ann D [ s ] f s − p ⊆ D [ s ] h V α i Ann D [ s ] f s − p ⊆ D [ s ]Ann D [ s ] f s − p . The set V α ∩ R [ s ] in Line 16 forms a basis of the R [ s ]-module W − α ∩ R [ s ] since one hasAnn D [ s ] f s ∩ R [ s ] = 0. It remains to show how and R -basis of G p V − α R [ f − ] is obtainedfrom W α ∩ R [ s ]. Given an elimination order < s for s in R [ s ], we have that R [ s ] < s = R < ′ s [ s ]where < ′ s is the monomial order induced by < s in R . Therefore, for any f ∈ R [ s ], if theleading monomial of f with respect to < s is in R · s i , then f ∈ L ij =0 R · s j . Consequently, a N ALGORITHM FOR HODGE IDEALS 9
Gr¨obner basis H α of V α ∩ R [ s ] with respect to < s gives a basis of V α ∩ R [ s ] as R -submoduleof L ij =0 R · s j .Taking the elements of degree at most p in s from H α and making the substitution s = − ∂ t t leaves us with elements of the form P pj =0 h ′ j ∂ jt t j . Line 17 denotes by H ( p ) α theset of such elements. The isomorphism D [ s ]Ann D [ s ] f s − p ∼ = D [ s ] f s − p ∼ = t − p D [ s ] f s ∼ = t − p D [ ∂ t t ] δ is given by sending the class of 1 to t − p δ . Therefore, since tδ = f δ , the claimed R -basisof G p V − α ι + R will be given by the set B ( p ) α obtained from H ( p ) α after substituting t = f and multiplying by f − p . Notice that the elements of B ( p ) α are well-defined in ι + R [ f − ].However, for α < B ( p ) α is actually in ι + R , that is, the division by f p in Line 18 giverise to no rational functions. Indeed, this will follow from Lemma 4.3 once we show that B ( p ) α ⊆ G p V − α ι + R [ f − ].Let α ′ be a rational number from the set in Line 14 such that α ′ < α . Then, wehave B ( p ) α ′ ⊆ B ( p ) α . If α ′ is the largest of such numbers, then ∂ t t + α acts nilpotently on R h B ( p ) α i /R h B ( p ) α ′ i by construction. It remains to show that t · R h B ( p ) α i ⊆ R h B ( p ) α +1 i and ∂ t · R h B ( p ) α i ⊆ R h B ( p +1) α − i . Let u ∈ P − α ∼ = D [ s ] h K α i / Ann D [ s ] f s − p , then the first inclusionfollows from the equality t · ( s − α ) n u = ( s − α + 1) n t · u = 0 . For the second one, notice that sW α ⊆ W α . Then, the second inclusion follows by theidentity ∂ t = t − ( ∂ t t − B ( p ) α ⊆ G p V − α ι + R .In order to conclude the proof, the last thing remaining is to show the reverse inclusion G p V − α ι + R ⊆ R h B ( p ) α i . But this follows from the fact that the isomorphism Ψ fromProposition 4.1 sends the elements of G p ι + R to T p ι + R [ f − ] ∼ = D [ s ] f s − p . (cid:3) An straightforward application of Theorem 3.1 gives the following corollary.
Corollary 7.2.
For any p ∈ N and f ∈ R reduced, Algorithm 1 computes generators forthe Hodge ideals I k ( f α ) , α ∈ Q ∩ (0 , , k = 0 , . . . , p . In contrast with G p V α ι + R [ f − ], assuming that p ∈ N is fixed, the set of ideals I p ( f α )for α ∈ Q ∩ (0 ,
1] is not finite since the Hodge ideals depend on α even if the V -filtrationdoes not, i.e. V α ι + R = V α + ǫ ι + R for 0 < ǫ ≪
1. To remedy this and still have a finiteoutput, one can work on a transcendental extension R ( α ) = C ( α )[ x , . . . , x n ] of the basefield and compute with the polynomials Q i ( α ) from Theorem 3.1 symbolically. Remark 3.1.
In practice, for a fixed α ∈ Q ∩ (0 , I k ( f α ) for k = 0 , . . . , l where l is the generating level of the Hodge filtration on M ( f − α ). Recall that the generating level of any D X -module ( M , F • ) equipped with agood filtration is the smallest integer l such that F k D X · F l M = F l + k M for all k ≥ . Therefore, in our case, for k > l , I k ( f α ) is generated by f · I k − ( f α ) and(7.4) { f D ( h ) − ( α + k ) hD ( f ) | h ∈ I k − ( f α ) , D ∈ Der C ( R ) } . see [MP19b, § f ∈ O X ( X ), the Hodge filtration on M ( f − α )is known to be generated at level n − ⌈ ˜ α f + α ⌉ , see [MP20a, Theorem E], where ˜ α f is theminimal exponent of f , that is, the smallest root of b f ( − s ) / (1 − s ). Conversely, noticethat from Corollary 7.2 and (7.4) one can compute the minimal generating level of theHodge filtration on M ( f − α ).After Theorem 3.2, we also obtain a new algorithm to compute the multiplier idealsand the (global) jumping numbers of any effective divisor, not necessarily reduced. Corollary 7.3.
For any f ∈ R , Algorithm 1 computes a set of generators of the multiplierideals J ( f α ) , α ∈ Q ∩ (0 , and the jumping numbers of f . Examples.
Let us show some non-trivial examples of Hodge ideals computed withAlgorithm 1.
Example 1.
Let f = x + y + x y . This is perhaps the simplest plane curve which isnot quasi-homogeneous or a µ -constant deformation of a quasi-homogeneous singularity. α I ( f α ) I ( f α ) R ( x , x y, xy , y ) R ( x , x y, xy , y )
12 (2) R (5 x + 2 xy , x y, x y , xy , y + 2 x y ) ( x, y ) ((5 α − x + (2 α − x y , x y, xy , (5 α − y + (2 α − x y ) ( x , xy, y ) ( x , x y, x y , x y , xy , y )1 (2) ( x , xy, y ) ( x − (2 α − x y , x y, x y , x y , x y , xy , y − (2 α − x y ) Table 1.
Hodge ideals I p ( f α ) , p = 0 , , α ∈ Q ∩ (0 ,
1] for f = x + y + x y .The subscripts in the first columns of Tables 1, 2 and 3 denote the nilpotency index of ∂ t t − α on Gr αV . Example 2.
Let f λ = ( y − x )( y + λx ) , λ ∈ C ∗ . The parameter λ ∈ C ∗ is an analyticalinvariant of the singularity defined by f λ at the origin. That is, for different values of λ ∈ C ∗ , the singularities f λ are not analytically equivalent. Set h := λx + ((2 α − λ − α + 1) x y + (4 α − y h := (( α − λ − λα + α − x y + ((2 α − λ − α + 1) y h := (( α − λ − λα + α − x y − (4 α − y h := x y + λx + ((2 α − λ − α + 1) x y + (4 α − x y N ALGORITHM FOR HODGE IDEALS 11 α I ( f αλ ) I ( f αλ ) R ( y , xy , x y, x ) R ( y , x y , x y, x ) R ( y , xy , x y , ( λ − x y + 2 y , x ) R ( y , xy , ( λ − x y + 2 y , λx − ( λ − x y ) R ( y , ( λ − x y + 2 y , x y , λx − ( λ − x y ) ( x, y ) ( y , xy , x y , ( λ − x y + 2 y , ( λ − x y + 2 xy , λx + y ) ( y, x ) ( y , x y , ( λ − x y + 2 xy , x y, h ) ( x , xy, y ) ( y , x y , x y , ( λ − x y + 2 xy , ( λ − x y + 2 x y , λx + xy ) ( y , xy, x ) ( y , x y , x y , x y , x y, xh ) ( y , x y, x ) ( y , xy , x y , h , x y , h , x )1 (2) ( y , xy , x y, x ) ( y , xy , x y , x y , x y , h ) Table 2.
Hodge ideals I p ( f αλ ) , p = 0 , , α ∈ Q ∩ (0 ,
1] for f = ( y − x )( y − λx ). Example 3.
Let f = x + y + z + xyz . Then, the pair ( A C , div( f )) is log-canonical.Therefore, the multiplier ideals are trivial. Hence, the Hodge ideals provide a first non-trivial invariant of the singularity. α I ( f α ) I ( f α ) I ( f α ) R ( x, y, z ) ( x y, x z, xy , zy , yz , xz , y − z , x − z , xyz + 3 z , z ) R ( x, y, z ) ( x, y, z ) (2) R ( z , yz , xz , ( z , yz , xz , y z , xyz , x z , y z , xy z , x yz ,xz + 3 y , (3 α + 56) x z + y z − αxyz − (81 α + 28) z ,xy + 3 z , αy + 18 αxy z + (3 α + 56) x z ,yz + 3 x ) 27 αxy + 9 αx yz + (81 α + 28) y z + xz , (3 α + 56) x y + 18 αxyz + 27 αz , αx y + 9 αxy z + (81 α + 28) x z + yz , αx + 18 αx yz + (3 α + 56) y z ) Table 3.
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