Divisors of expected Jacobian type
aa r X i v : . [ m a t h . A C ] A p r DIVISORS OF EXPECTED JACOBIAN TYPE
JOSEP `ALVAREZ MONTANER AND FRANCESC PLANAS-VILANOVA
Abstract.
Divisors whose Jacobian ideal is of linear type have received a lot of attentionrecently because of its connections with the theory of D -modules. In this work we are interestedon divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear typeand the relation type of its Jacobian ideal coincides with the reduction number with respectto the gradient ideal plus one. We provide conditions in order to be able to describe preciselythe equations of the Rees algebra of the Jacobian ideal. We also relate the relation type ofthe Jacobian ideal to some D -module theoretic invariant given by the degree of the Kashiwaraoperator. Introduction
Let (
X, O ) be a germ of a smooth n -dimensional complex variety and O X,O the ring of germsof holomorphic functions in a neighbourhood of O , which we identify with R = C { x , . . . , x n } by taking local coordinates. Let D R [ s ] be the polynomial ring in an indeterminate s withcoefficients in the ring of differential operators D R = R h ∂ , . . . , ∂ n i where ∂ i are the partialderivatives with respect to the variables x i . To any hypersurface defined by f ∈ R we mayattach several invariants coming from the theory of D -modules that measure its singularities.The goal of this work is to get more insight on the parametric annihilator Ann D R [ s ] ( f s ) := { P ( s ) ∈ D R [ s ] | P ( s ) · f s = 0 } , where we understand f s as a formal symbol that takes theobvious meaning f r when specializing to any integer r ∈ Z . This is the defining ideal ofthe D R [ s ]-modules generated by f s , that we denote as D R [ s ] f s , which plays a key role inthe theory of Bernstein-Sato polynomials as shown by Kashiwara [14]. Among the differentialoperators annihilating f s there exists the so-called Kashiwara operator [14, Theorem 6.3] thathas been used in some of the first algorithmic approaches to the computation of Bernstein-Sato polynomials given by Yano [29] and Brian¸con et al. [5]. Furthermore, the degree of theKashiwara operator is an interesting analytic invariant of the singularity, although it is muchcoarser than the Bernstein-Sato polynomial itself.A common theme in the study of the parametric annihilator is whether it is generated byoperators of degree one. This and some related linearity properties have been used by severalauthors in a wide range of different problems [7], [26], [20], [8] , [2], [21], [27], [28]. This linearityproperty of differential operators can be checked using algebraic methods as it was proved byCalder´on-Moreno and Narv´ez-Macarro in [7]. Namely, this property holds whenever the Jacobianideal of the hypersurface f is of linear type, that is the Rees algebra and the symmetric algebraof the Jacobian ideal coincide. Date : April 21, 2020.Both authors are supported by the Spanish Ministerio de Econom´ıa y Competitividad MTM2015-69135-Pand Generalitat de Catalunya SGR2017-932.
The aim of this paper is to get further connections between the Rees algebra of the Jacobianideal and the parametric annihilator. Building upon work of Mui˜nos and the second authorin [17], we introduce in Definition 4.2 the notion of divisors of expected Jacobian type as thosedivisors whose gradient ideal is of linear type and whose Jacobian ideal has relation type equalto its reduction number plus one (see also Definition 3.3). In Remark 4.3, it is easily seen thatthis is a natural generalization of the divisors of linear Jacobian type considered in [7] (see also[20]). For divisors of linear Jacobian type we can find an equation that resembles the initialterm or symbol of the Kashiwara operator with respect to a given order, and indeed this is thecase under some extra conditions.The organization of the paper is as follows: in Section 2 we review the basics on the equationsof Rees algebras and recover and extend some of the results of Mui˜nos and the second author in[17]. Section 3 and 4 are devoted to introduce the notion of ideal of expected relation type andits specialization to the case of the Jacobian ideal of a hypersurface. In Section 5 we describethe connection between divisors of expected Jacobian type and the parametric annihilator. Werelate the degree of the Kashiwara operator with the relation type of the Jacobian ideal inProposition 5.3. In Section 6 we present several examples in which we explore the case in whichthe ideal has the expected relation type. We also study some cases in which this condition isnot satisfied.Any unexplained notation or definition can be found in [6] or [25]. Throghout the paper,( R, m ) is a Noetherian local ring and b ⊆ a and J ⊆ I are ideals of R . Acknowledgements:
This work grew up from early conversations with Ferran Mui˜nos andwe are really grateful for his insight. We would also thank Jos´e Mar´ıa Giral for some helpfulcomments. Part of this work was done during a research stay of the first author at CIMAT,Guanajuato with a Salvador de Maradiaga grant (ref. PRX 19/00405) from the Ministerio deCiencia, Innovaci´on y Universidades.2.
On the equations of Rees algebras
Let ( R, m ) be a Noetherian local ring and let a = ( f , . . . , f m ) be an ideal of R , m ≥ R ( a ) = R [ a t ] = L d ≥ a d t d ⊂ R [ t ] be the Rees algebra of a . Let A = R [ ξ , . . . , ξ m ] bea polynomial ring in a set of variables ξ , . . . , ξ m and coefficients in R . Consider the gradedsurjective morphism ϕ : A → R ( a ) sending ξ i to f i t , for i = 1 , . . . , n + 1. The kernel of thismorphism is a graded ideal Q = L d ≥ Q d , whose elements will be referred to as the equationsof R ( a ). Let Q h d i be the ideal generated by the homogeneous equations of degree at most d .We then have an increasing sequence Q h i ⊆ Q h i ⊆ · · · ⊆ Q that stabilizes at some point. Thesmallest integer L ≥ Q h L i = Q is the relation type of R ( a ) and will be denotedrt( a ). We say that a is an ideal of linear type when rt( a ) = 1.Observe that the ideal Q depends on the polynomial presentation ϕ . Nevertheless, thequotients ( Q/Q h d − i ) d , for d ≥
2, do not (see [23]). Indeed, let α : S ( a ) → R ( a ) be thecanonical graded surjective morphism between the symmetric algebra S ( a ) of a and the Reesalgebra R ( a ) of a . Given d ≥
2, the d -th module of effective relations of a is defined to be E ( a ) d = ker( α d ) / a · ker( α d − ). One shows that, for d ≥ E ( a ) d ∼ = ( Q/Q h d − i ) d . In particular,the relation type of a can be calculated as the least integer L ≥
1, such that E ( a ) d = 0, for all d ≥ L + 1. Moreover, it is known that E ( a ) d ∼ = H ( f t, . . . , f m t ; R ( a )) d , where the right-hand IVISORS OF EXPECTED JACOBIAN TYPE 3 module stands for the degree d -component of the first Koszul homology module associated tothe sequence of degree one elements f t, . . . , f m t of R ( a ) ([23, Theorem 2.4]).The characterization of E ( a ) d in terms of the Koszul homology was used in [17] in order toobtain the equations of R ( a ) for equimultiple ideals a of deviation one. Our purpose in thissection is to rephrase, and extend a little bit, some of those results, but doing more emphasis inthe Koszul conditions than in the “regular sequence type conditions”. These characterizationswill be applied in the next sections to the Jacobian ideal of a hypersurface. For the sake ofcompleteness and self-containment, we outline parts of the line of reasoning in [17]. Let us startby fixing our general notations. Setting 2.1.
Let ( R, m ) be a Noetherian local ring, n ≥
2. Let f , . . . , f n ∈ m and f = f n +1 ∈ m .Let J = ( f , . . . , f n ) and I = ( f , . . . , f n , f ) = ( J, f ) be ideals of R . For i = 1 , . . . , n + 1, let J i = ( f , . . . , f i ); set J = 0 and observe that J n = J and J n +1 = I .For i = 1 , . . . , n + 1 and d ≥ , set T i,d = ( J i − I d − : f i ) ∩ I d − J i − I d − . For d = 1 , set T i, = J i − : f i . Note that for i = 1 (and any d ≥ T ,d = (0 : f ) ∩ I d − . For i = n + 1 and d ≥
2, it wasshown in [17, Proof of Lemma 3.1] that: T n +1 ,d = ( J I d − : f ) ∩ I d − J I d − ∼ = J I d − : f d J I d − : f d − . (2.1)The isomorphism goes as follows. Given a ∈ ( J I d − : f ) ∩ I d − , since I d − = J I d − + f d − R ,write a = b + cf d − with b ∈ J I d − and c ∈ R . The class of an element a ∈ ( J I d − : f ) ∩ I d − is sent to the class of c ∈ J I d − : f d . Notation 2.2.
A graded Koszul complex.
Let us denote K ( z , . . . , z r ; U ) the Koszulcomplex of a sequence of elements z , . . . , z r of a ring U . Since U will always be the Reesalgebra R ( I ) of I , we just skip the letter U . For i = 1 , . . . , n + 1, we consider the sequencesf i t := f t, . . . , f i t of elements of degree one in R ( I ); we highlight the distinct notation with thelength one sequence f i t . Set ft := f n+1 t = f t, . . . , f n t, f t . Thus K (f i t) = K ( f t, . . . , f i t ; R ( I ))stands for the Koszul complex associated to f i t = f t, . . . , f i t , with first nonzero zero terms: K (f i t) : . . . → K (f i t) ∂ −→ K (f i t) ∂ −→ K (f i t) → . Let H j (f i t) = H j ( K (f i t)) be its j -th homology module. Note that, since R ( I ) is a graded algebra, K (f i t), and hence its homology, inherit a natural grading. The first nonzero terms of the degree d -component K (f i t) d , d ≥
2, (omitting the powers of the variable t ) are: . . . → K (f i t) d = ∧ ( R i ) ⊗ I d − ∂ ,d − −→ K (f i t) d = ∧ ( R i ) ⊗ I d − ∂ ,d − −→ K (f i t) d = I d → . The Koszul differentials are defined as follows: if e , . . . , e i stands for the canonical basis of R i and u ∈ I d − and v ∈ I d − , then ∂ ,d − ( e j ∧ e l ⊗ u ) = e l ⊗ f j u − e j ⊗ f l u and ∂ ,d − ( e j ⊗ v ) = f j v. Note that, under the isomorphism ∧ ( R i ) ⊗ I d − ∼ = I d − ⊕ ( i ) · · · ⊕ I d − , the differential ∂ ,d − sends the i -th tuple ( a , . . . , a i ) ∈ ( I d − ) ⊕ i to the element a f + · · · + a i f i ∈ I d . In particular, J. `ALVAREZ MONTANER AND F. PLANAS-VILANOVA for d = 1, H (f i t) = { ( a , . . . , a i ) ∈ R i | P ij a j f j = 0 } = Z ( f , . . . , f i ), the first module ofsyzygies of J i = ( f , . . . , f i ). Remark 2.3.
Equations vs cycles.
Let Q be the ideal of equations of R ( I ). As said before, E ( I ) d ∼ = (cid:18) QQ h d − i (cid:19) d ∼ = H (ft) d = H ( K (ft)) d = H ( f t, . . . , f n t, f t ; R ( I )) d , (2.2)i.e., the d -th module of effective relations E ( I ) d of I is isomorphic to the degree d -component ofthe first Koszul homology module H (ft) of ft, where d ≥
2. This isomorphism sends the classof an equation P ∈ Q d to the class of the cycle ( P (f) , . . . , P n (f) , P n +1 (f)) ∈ L n +1 j =1 I d − , wheref = f , . . . , f n , f , P = P n +1 j =1 ξ j P j , and P j ∈ A d − = R [ ξ , . . . , ξ n +1 ] d − . (See [17, Remark 2.1].)The next two remarks are devoted to write more explicitly some complexes and morphismsthat well be used subsequently. Remark 2.4.
A short exact sequence of Koszul complexes.
Let K ( f i t ) be the Koszulcomplex associated to the length one sequence f i t ∈ R ( I ). So K ( f i t ) = R ( I ), K ( f i t ) = ∧ ( R ) ⊗ R ( I ) ∼ = R ( I ), and K j ( f i t ) = 0, for j = 0 ,
1. In degree d ≥ K ( f i t ) d = I d , K ( f i t ) = ( ∧ ( R ) ⊗ R ( I )) d ∼ = I d − and, for a ∈ I d − , then ∂ ,d − ( a ) = af i .There is an isomorphism of Koszul complexes K (f i t) ∼ = K (f i − t) ⊗ K ( f i t ). Concretely, K p (f i t) ∼ = M r + s = p K r (f i − t) ⊗ K s ( f i t ) = K p (f i − t) ⊗ K ( f i t ) ⊕ K p − (f i − t) ⊗ K ( f i t ) ∼ = K p (f i − t) ⊗ R ( I ) ⊕ K p − (f i − t) ⊗ R ( I ) ∼ = K p (f i − t) ⊕ K p − (f i − t) , which induces a short exact sequence of Koszul complexes:0 → K (f i − t) → K (f i t) → K (f i − t)( − → , (2.3)where K (f i − t)( −
1) is the shifted complex by -1, i.e., K s (f i − t)( −
1) = K s − (f i − t). In particular,for d ≥
1, the degree d -component gives rise to the the short exact sequence of complexes:0 → K (f i − t) d → K (f i t) d → K (f i − t)( − d → . Displaying by columns the first nonzero terms of each complex, we get:... (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) / / ∧ ( R i − ) ⊗ I d − / / (cid:15) (cid:15) ∧ ( R i ) ⊗ I d − / / (cid:15) (cid:15) R i − ⊗ I d − / / (cid:15) (cid:15) / / R i − ⊗ I d − / / (cid:15) (cid:15) R i ⊗ I d − / / (cid:15) (cid:15) I d − / / (cid:15) (cid:15) / / I d / / (cid:15) (cid:15) I d / / (cid:15) (cid:15) / / (cid:15) (cid:15)
00 0 0 . IVISORS OF EXPECTED JACOBIAN TYPE 5
The middle row, 0 → K (f i − t) d → K (f i t) → K (f i − t)( − d →
0, is nothing else than:0 → I d − ⊕ ( i − · · · ⊕ I d − −→ I d − ⊕ ( i ) · · · ⊕ I d − → I d − → , where the first morphism sends ( a , . . . , a i − ) to ( a , . . . , a i − , a , . . . , a i ) to a i , the projection to the last component. Remark 2.5.
The long exact sequence in homology.
In turn, the short exact sequence(2.3) induces the long exact sequence in homology. We display its degree d -component, d ≥ . . . → H ( K (f i − t)( − d δ −→ H (f i − t) d → H (f i t) d → H ( K (f i − t)( − d δ −→→ H (f i − t) d → H (f i t) → H ( K (f i − t)( − d → . Clearly, H j ( K (f i − t)( − d = H j − (f i − t) d − and H ( K (f i − t)( − d = 0. The connectingmorphism is known to be the multiplication by the element ± f i t . Thus we get: . . . → H (f i − t) d − ·± f i t −→ H (f i − t) d → H (f i t) d →→ H (f i − t) d − ·± f i t −→ H (f i − t) d → H (f i t) d → . If d = 1, then H (f i − t) = 0, H (f i − t) = R and H (f i − t) = I/J i − . Henceker (cid:16) H (f i − t) ·± f i t −→ H (f i − t) (cid:17) = ( J i − : f i ) = T i, . In particular, for i = 1 , . . . , n + 1 and d = 1, one deduces the exact sequence:0 → H (f i − t) → H (f i t) → T i, → , (2.4)where H (f i − t) = Z ( f , . . . , f i − ) and H (f i t) = Z ( f , . . . , f i ).If d ≥
2, one can check that H (f i − t) d − = I d − /J i − I d − and H (f i − t) d = I d /J i − I d − .Thus ker (cid:16) H (f i − t) d − ·± f i t −→ H (f i − t) d (cid:17) = ( J i − I d − : f i ) ∩ I d − /J i − I d − = T i,d . (See Setting 2.1.) In particular, for i = 1 , . . . , n + 1 and d ≥
2, we deduce the exact sequence: H (f i − t) d − ·± f i t −→ H (f i − t) d → H (f i t) d → T i,d → . (2.5)Note that the middle morphism in (2.5) is induced by the inclusion. Namely, the class of acycle ( a , . . . , a i − ), a j ∈ I d − , maps to the class of the cycle ( a , . . . , a i − , a , . . . , a i ) a i .We recover [17, Lemma 3.1]. Keeping the notations as in Setting 2.1 and Notation 2.2: Corollary 2.6.
For d ≥ , the following sequence is exact. → H (f n t) d f t · H (f n t) d − −→ E ( I ) d −→ J I d − : f d J I d − : f d − → . (2.6) The right-hand morphism sends the class of an equation P ∈ Q d to the class of P (0 , . . . , , . J. `ALVAREZ MONTANER AND F. PLANAS-VILANOVA
Proof.
Take i = n + 1 and d ≥ H (f n+1 t) d = H (ft) d , which by (2.2), isisomorphic to E ( I ) d . See also the definition of T n +1 ,d and its isomorphic expression in (2.1).The second part follows from the composition of the morpshims in (2.2), (2.5) and (2.1). Indeed,the class of P ∈ Q d is sent to the class of ( P (f) , . . . , P n +1 (f)) ∈ L n +1 j =1 I d − through (2.2), where P = P n +1 j =1 ξ j P j , P j ∈ A d − . By (2.5), ( P (f) , . . . , P n +1 (f)) is sent to the class of P n +1 (f) ∈ ( J I d − : f ) ∩ I d − . Write P n +1 = P nj =1 ξ j Q j + cξ d − n +1 , with Q j ∈ A d − and c ∈ R . In particular, P n +1 (f) = b + cf d − , with b = P nj =1 f j Q j (f) ∈ J I d − . Then the isomorphism (2.1) sends the classof P n +1 (f) to the class of c ∈ J I d − : f d . Observe that P (0 , . . . , ,
1) = P n +1 (0 , . . . , ,
1) = c . (cid:3) The first part of the following result is shown in [17, Lema 3.3]. Our proof here is a directconsequence of Remarks 2.4 and 2.5, and the sequences (2.4) and (2.5). We keep the notationsas in Setting 2.1 and Notation 2.2.
Theorem 2.7.
Fix d ≥ and i = 1 , . . . , n + 1 . ( a ) The following two conditions are equivalent: ( i ) H ( f t ) d = 0 , H (f t) d = 0 , . . . , H (f i t) d = 0 ; ( ii ) T ,d = 0 , T ,d = 0 , . . . , T i,d = 0 . ( b ) Suppose that, for some i = 1 , . . . , n , T ,d = 0 , T ,d = 0 , . . . , T i,d = 0 . Then H (f i+1 t) d ∼ = T i +1 ,d . ( c ) Fix now d ≥ . Suppose that, for some i = 1 , . . . , n − , T ,d = 0 , T ,d = 0 , . . . , T i,d = 0 and that T ,d − = 0 , T ,d − = 0 , . . . , T i,d − = 0 . Then the following sequence is exact. → ( J i I d − : f i +1 ) ∩ I d − f i +2 · [( J i I d − : f i +1 ) ∩ I d − ] + J i I d − → H (f i+2 t) d → T i +2 ,d → . (2.7) Proof.
The implication ( i ) ⇒ ( ii ) follows directly from the exact sequences (2.4) and (2.5). Since H (f t) = 0, then, by (2.4) and (2.5), H ( f t ) d ∼ = T ,d and ( ii ) ⇒ ( i ) holds for i = 1. Supposethat ( ii ) ⇒ ( i ) holds for i − ≥
1. By the induction hypothesis, H (f i − t) d = 0 and, by (2.4)and (2.5), H (f i t) d ∼ = T i,d = 0. This proves ( a ).Suppose now that, for some i = 1 , . . . , n − T ,d = 0 , T ,d = 0 , . . . , T i,d = 0. In particular,since ( ii ) ⇒ ( i ), H ( f t ) d = 0 , H (f t) d = 0 , . . . , H (f i t) d = 0. Using (2.4) and (2.5), for theinteger i + 1, H (f i+1 t) d ∼ = T i +1 ,d . This proves ( b ).Suppose now that the hypotheses in ( c ) hold. Then, by ( b ) applied to d −
1, we obtain theisomorphism H (f i+1 t) d − ∼ = T i +1 ,d − . Therefore, H (f i+1 t) d − ∼ = ( J i I d − : f i +1 ) ∩ I d − J i I d − and H (f i+1 t) d ∼ = ( J i I d − : f i +1 ) ∩ I d − J i I d − . Through these isomorphisms, H (f i+1 t) d f i +2 t · H (f i+1 t) d − ∼ = ( J i I d − : f i +1 ) ∩ I d − f i +2 · [( J i I d − : f i +1 ) ∩ I d − ] + J i I d − . The rest follows from the exact sequence (2.5) applied to i + 2. (cid:3) IVISORS OF EXPECTED JACOBIAN TYPE 7
Next we specialise Theorem 2.7 ( c ), to the case n = 2. Corollary 2.8.
Let ( R, m ) be a Noetherian local ring, f , f , f ∈ m and let J = ( f , f ) and I = ( f , f , f ) . Fix d ≥ . Assume that (0 : f ) ∩ I d − = 0 and (0 : f ) ∩ I d − = 0 . Then thefollowing sequence is exact. → ( f I d − : f ) ∩ I d − f · [( f I d − : f ) ∩ I d − ] + f I d − −→ E ( I ) d −→ ( J I d − : f d )( J I d − : f d − ) → . (2.8) Proof.
Take n = 2, i = 1 and d ≥ c ). Then T ,d − = (0 : f ) ∩ I d − and T ,d = (0 : f ) ∩ I d − , which are zero by hypothesis. The rest follows from the sequence (2.7). (cid:3) Corollary 2.9.
Let ( R, m ) be a Noetherian local ring, f , f , f ∈ m and let J = ( f , f ) and I = ( f , f , f ) . Fix L ≥ . Suppose that (0 : f ) ∩ I d − = 0 , for all ≤ d ≤ L , and that ( f : f ) ⊆ ( f : f ) . Then the following two conditions are equivalent. ( a ) T ,d = ( f I d − : f ) ∩ I d − /f I d − = 0 , for all ≤ d ≤ L ; ( b ) E ( I ) d ∼ = ( J I d − : f ) / ( J I d − : f d − ) , for all ≤ d ≤ L .Proof. The hypotheses (0 : f ) ∩ I d − = 0, for all 1 ≤ d ≤ L , allows us to apply Corollary 2.8,for all 2 ≤ d ≤ L .Suppose that ( a ) holds. The vanishing of T ,d ensures the vanishing of the left-hand side termin the exact sequence (2.8). Thus ( b ) holds.Conversely, assume that ( b ) holds. Let us prove T ,d = 0, by induction on d , 2 ≤ d ≤ L . Sotake d = 2. Using ( b ) and (2.8), for d = 2, then ( f I : f ) ∩ I = f · ( f : f ). Using the hypothesis( f : f ) ⊆ ( f : f ), we get( f I : f ) ∩ I = f · ( f : f ) ⊆ f · ( f : f ) ⊆ ( f ) . Thus T , = 0. Take now d ≥ d ≤ L . By the induction hypothesis T ,d − = 0, so( f I d − : f ) ∩ I d − = f I d − . Using ( b ) and (2.8), for such d , then( f I d − : f ) ∩ I d − = f · [( f I d − : f ) ∩ I d − ] + f I d − = f · [ f I d − ∩ I d − ] + f I d − ⊆ f I d − . Hence T ,d = 0. (cid:3) Remark 2.10.
In the case that f , f is a regular sequence, then clearly (0 : f ) ∩ I d − = 0 forall 1 ≤ d ≤ L and ( f : f ) = f R ⊆ ( f : f ). However the converse does not always holds as thenext example shows. Example 2.11.
Let R = k [[ x, y ]] be the formal power series ring in two variables over a field k of characteristic zero. Take a, b ≥ J = ( f , f ) and I = ( f , f , f )with f = x a y b , f = dfdx = ax a − y b and f = dfdy = bx a y b − . Then ( f : f ) = yR ⊆ ( f : f ) = R, whereas f , f is not a regular sequence. We point out that J = I is an ideal of linear type. J. `ALVAREZ MONTANER AND F. PLANAS-VILANOVA Ideals with expected relation type
We recall now a central concept to our purposes.
Definition 3.1.
Let ( R, m ) be a Noetherian local ring and let a and b be two ideals of R . Theideal b is a reduction of a if b ⊆ a and there is an integer r ≥ a r +1 = ba r . From thedefinition it follows that rad ( b ) = rad ( a ), Min( R/ b ) = Min( R/ a ) and height ( b ) = height ( a )(see, e.g.,[25, Lemma 8.10]). Note that the ideal a is always a reduction of itself. An ideal a which has no reduction other than itself is called a basic ideal . The smallest integer r ≥ a r +1 = ba r is called the reduction number of a with respect to b and is denoted rn b ( a ). For b = a , rn b ( a ) = 0. (see [22]).The next result is shown in [17, Lemma 3.1]. We deduce it here from our previous remarks. Proposition 3.2.
Let ( R, m ) be Noetherian local ring, n ≥ , and let J = ( f , . . . , f n ) be areduction of I = ( f , . . . , f n , f ) . Then rn J ( I ) + 1 ≤ rt( I ) . Proof.
Let rt( I ) = L ≥
1. Hence, E ( I ) d = 0, for all d ≥ L + 1. By the exact sequence (2.6), T n +1 ,d = 0, for all d ≥ n + 1. Therefore ( J I d − : f d ) = ( J I d − : f d − ), for all d ≥ L + 1. Since J is a reduction of I , then ( J I m − : f m ) = R , for m ≫ f L ∈ J I L − , I L = J I L − and rn J ( I ) ≤ L − (cid:3) Definition 3.3.
Let ( R, m ) be a Noetherian local ring and let J = ( f , . . . , f n ) be a reductionof I = ( f , . . . , f n , f ). We say that I has the expected relation type with respect to J ifrn J ( I ) + 1 = rt( I ) . When J is understood by the context, we will skip the locution “with respect to J”. Example 3.4.
Let ( R, m , k ) be a Noetherian local ring with k = R/ m an infinite field. Let a be an ideal of R .( a ) If a is of linear type, then a is basic and has the expected relation type.( b ) If a is a parametric ideal, that is, generated by system of parameters, then a is basic,but it is not necessarily of linear type, nor it has necessarily the expected relation type. Proof.
Let b be a reduction of a . By [22, 2. Theorem 1], there exists an ideal c ⊆ b ⊆ a , whichis a minimal reduction of a , that is, no ideal strictly contained in c is a reduction of a . By [22,2. Lemma 3], every minimal set of generators of c = ( x , . . . , x s ) can be extended to a minimalset of generators of a = ( x , . . . , x s , x s +1 , . . . , x m ), with µ ( c ) = s ≤ µ ( a ) = m , where µ ( · ) standsfor the minimal number of generators. If a is of linear type, then S ( a ) ∼ = R ( a ). On tensoring by A/ m , k [ T , . . . , T m ] ∼ = F ( a ), where F ( a ) = ⊕ d ≥ a d / ma d is the fiber cone of a . On taking Krulldimensions, we get µ ( a ) = m = l ( a ), where l ( a ) = dim F ( a ) is the analytic spread of a . By [6,Proposition 4.5.8], l ( a ) ≤ µ ( c ). Thus s = m and c = a . Therefore b = a and a is basic. Inparticular, rn b ( a ) = 0 and, since a is of linear type, rt( a ) = 1 = rn b ( a ) + 1. This proves ( a ).If a is a parametric ideal, then height ( a ) = µ ( a ). By [22, 4. Theorem 5], a is basic. Takenow R = k [[ x, y, z, w ]], where w = wz = 0, and a = ( x m − y + z m , x m , y m ), m ≥
2. Then a is aparameter ideal, hence a basic ideal, but its relation type is at least m (see [1, Example 2.1]). (cid:3) The following result gives a characterization of ideals with expected relation type in terms ofthe Koszul homology.
IVISORS OF EXPECTED JACOBIAN TYPE 9
Proposition 3.5.
Let ( R, m ) be a Noetherian local ring and let J = ( f , . . . , f n ) be a reductionof I = ( f , . . . , f n , f ) . The following conditions are equivalent. ( a ) I has the expected relation type; ( b ) H (f n t) d = f t · H (f n t) d − , for all d ≥ rn J ( I ) + 2 .In particular, if T i,d = ( J i − I d − : f i ) ∩ I d − /J i − I d − = 0 , for all d ≥ rn J ( I ) + 2 and all i = 1 , . . . , n , then I has the expected relation type.Proof. Set r = rn J ( I ). If I has the expected relation type, then E ( I ) d = 0, for all d ≥ r + 2.In particular, using the exact sequence (2.6), we deduce H (f n t) d = f t · H (f n t) d − , for all d ≥ r + 2. Conversely, if H (f n t) d = f t · H (f n t) d − , for all d ≥ r + 2, then by (2.6), E ( I ) d ∼ =( J I d − : f d ) / ( J I d − : f d − ). However, if d ≥ r + 2, then f d − ∈ J I d − , so ( J I d − : f d − ) = R and E ( I ) d = 0. Therefore, rt( I ) ≤ r + 1 = rn J ( I ) + 1. The other inequality follows fromProposition 3.2. This shows the equivalence ( a ) ⇔ ( b ).If T ,d = , T ,d = 0 , . . . , T n,d = 0, for all d ≥ rn J ( I ) + 2, by Theorem 2.7, H (f n t) d = 0, for all d ≥ rn J ( I ) + 2, and, by ( b ) ⇒ ( a ), I has the expected relation type. (cid:3) The main result of Mui˜nos and the second author in [17], gives an instance of ideals of expectedrelation type. We rephrased it here in terms of our T i,d . Corollary 3.6.
Let ( R, m ) be a Noetherian local ring, n ≥ . Let f , . . . , f n ∈ m and f ∈ m .Let J = ( f , . . . , f n ) and I = ( f , . . . , f n , f ) be ideals of R . Assume that for all d ≥ and all i = 1 , . . . , n , T i,d = ( J i − I d − : f i ) ∩ I d − /J i − I d − = 0 . Then, for all d ≥ , E ( I ) d ∼ = ( J I d − : f d )( J I d − : f d − ) . In particular, if J is a reduction of I , then I has the expected relation type with respect to J .Proof. If T ,d = 0 , T ,d = 0 , . . . , T n,d = 0, for all d ≥
2, then, by Theorem 2.7 ( a ), H (f n t) d = 0,for all d ≥
2. By the exact sequence (2.6), E ( I ) d ∼ = ( J I d − : f d ) / ( J I d − : f d − ). The secondassertion follows directly from Proposition 3.5. (cid:3) Divisors of expected Jacobian type
Let (
X, O ) be a germ of a smooth n -dimensional complex variety and O X,O the ring of germsof holomorphic functions in a neighbourhood of O , which we identify with R = C { x , . . . , x n } by taking local coordinates. Let ( D, O ) be a germ of divisor defined locally by f ∈ R and set f i = dfdx i for i = 1 . . . , n . From now on, until the end of the paper, we consider the followingnotations. Setting 4.1.
Let R = C { x , . . . , x n } be the convergent power series ring, which is a Noetherianregular local ring (see, e.g., [25, Lemma 7.1]). Let f ∈ m . Set f i = dfdx i , for i = 1 . . . , n .Let J = ( f , . . . , f n ) and I = ( f , . . . , f n , f ) = ( J, f ). Note that if f ∈ m , then f i ∈ m and J ⊆ I ⊆ m . The ideals J and I are called the gradient ideal of f and the Jacobian ideal of f ,respectively. It is known that, when f ∈ m , then f ∈ ( x f , . . . , x n f n ) ⊆ J , where H stands forthe integral closure of the ideal H (see [25, Corollary 7.1.4]). In particular, J is a reduction of I (see, e.g., [25, Proposition 1.1.7]). Definition 4.2.
A germ of divisor (
D, O ), with reduced equation given by f , is of linear Jacobiantype if I is an ideal of linear type ([8, Definition 1.11]); f will be said of expected Jacobian type ,if J is of linear type and I has the expected relation type with respect to J . Remark 4.3.
Divisors of linear Jacobian type are divisors of expected Jacobian type. Indeed,by Example 3.4, ( a ), if f is of linear Jacobian type, then I is basic, thus J = I is of linear typeand I has the expected relation type since rt( I ) = 1 and rn J ( I ) = 0. Example 4.4.
Let R = C { x, y } be the convergent power series ring in two variables x, y . Let f ∈ C [ x, y ] be a polynomial such that f C [ λx + y ] , C [ x + µy ], for all λ, µ ∈ C . Then J is anideal of linear type minimally generated by two elements. Proof.
Since R is local and J = ( f , f ), to see that J is minimally generated by two elementsit is enough to prove that f ( f ) or f ( f ). Let us see that if f ∈ ( f ), then f is eitherin C [ y ], or else in C [ x + µy ], for some µ ∈ C (similarly, one would do the same if f ∈ ( f )).Since f C [ y ], deg x ( f ) = r ≥
1. Write f = P ri =0 x r g i ( y ) and suppose that f = pf , for someelement p ∈ C { x, y } . Since f and f are polynomials, then p ∈ C [ x, y ] must be a polynomialtoo. Equating the highest degree terms in x in the expression f = pf , one deduces that either p = 0, or else g ′ r ( y ) = 0. However, if p = 0, then f = 0 and f ∈ C [ y ], a contradiction. Thus g ′ r ( y ) = 0 and g r ( y ) = a r ∈ C , a r = 0, because deg x ( f ) = r ≥
1. Substituting g r ( y ) = a r in f andequating again the highest degree term in x in the equality f = pf , one gets pg ′ r − ( y ) = ra r = 0.Thus p ∈ C , p = 0. Setting µ = 1 /p , we have g r − ( y ) = ra r µy + a r − . Again, substitutingthis expression in f and equating the r − f = (1 /µ ) f , onegets g r − ( y ) = a r (cid:0) r (cid:1) ( µy ) + a r − µy + a r − . Proceeding recursively, one would get the equality f = P ri =0 a i ( x + µy ) i and so f would be an element of C [ x + µy ], a contradiction.Therefore, J is minimally generated by two elements. Now apply [13, Proposition 1.5]. Thus J can be generated by two elements which form a d -sequence. In particular J is of linear type(see, e.g., [25, Corollary 5.5.5]). (cid:3) Example 4.5.
Suppose that J = ( f , . . . , f n ) is generated by an R -regular sequence, for in-stance, if f has an isolated singularity at O (see, e.g., [24, IV, Remark 2.5]). In particular, J is of linear type ([25, Corollary 5.5.5]). There are three possibilities according to the previousdefinition. If I is of linear type, then f is a divisor of linear Jacobian type. Suppose that I isnot of linear type. Recall that, by the argument in Setting 4.1, J is a reduction of I and, byProposition 3.2, rn J ( I ) + 1 ≤ rt( I ). If the equality holds, then f is of expected Jacobian type.The third and last case occurs when J is of linear type, but rn J ( I ) + 1 < rt( I ), i.e., f is not ofexpected Jacobian type. Remark 4.6.
The linear type condition for the Jacobian ideal was investigated by Calder-´on-Moreno and Narv´aez-Macarro in [7, 8] (see also [20]). They proved that divisors of linearJacobian type are Euler homogeneous. Recall that a divisor D is Euler homogeneous if there isa vector field χ at O such that χ ( f ) = f , or in other words, f ∈ J . In particular, J = I andrn J ( I ) = 0.For divisors satisfying the conditions T i,d = 0 we can explicitly describe the equations of theRees algebra of the Jacobian ideal which corresponds to describe the blow-up at the singularlocus of the divisor. More precisely, and summarizing several results in a unique statement: IVISORS OF EXPECTED JACOBIAN TYPE 11
Theorem 4.7.
Let R = C { x , . . . , x n } be the convergent power series ring Let f ∈ m , J and I be as in Setting 4.1. Suppose that T i,d = 0 , for all d ≥ and all i = 1 , . . . , n . Set ξ i +1 = s . Let ϕ : R [ ξ , . . . , x n , s ] = C { x , . . . , x n } [ ξ , . . . , x n , s ] → R ( I )(4.1) be a polynomial presentation of R ( I ) , sending ξ i to f i t and s to f t . Let Q = L d ≥ Q d the idealof equations of R ( I ) . The following conditions hold. ( a ) f , . . . , f n is an R -regular sequence and J is of linear type. ( b ) J is a reduction of I and rt( I ) = rn J ( I ) + 1 . If f ∈ J , then f is a divisor of linearJacobian type; otherwise, f is a divisor of expected Jacobian type. ( c ) For all d ≥ , E ( I ) d ∼ = ( Q/Q h d − i ) d ∼ = ( J I d − : f d )( J I d − : f d − ) ;(4.2) the class of P ( ξ , . . . , ξ n , s ) ∈ Q d is sent to the class of P (0 , . . . , , ∈ ( J I d − : f d ) . ( d ) Set L = rt( I ) . A minimal generating set of equations of R ( I ) can be obtanied froma minimal generating set of Q , the first syzygies of I , and representatives of inverseimages of a minimal generating set of ( J I d − : f d ) / ( J I d − : f d − ) , for all ≤ d ≤ L . ( e ) There exists a unique top-degree equation of degree L , which is of the form (4.3) s L + p s L − + · · · + p L , where p j ∈ R [ ξ , . . . , ξ n ] are either zero, or else, polynomials of degree j .Proof. Since f ∈ m , then f i ∈ m and so J ⊆ I ⊆ m . Recall that T i, = ( J i − : f i ). There-fore, T i, = 0, for all i = 1 , . . . , n is equivalent to f , . . . , f n being an R -regular sequence. Inparticular, I is of linear type. This proves ( a ); ( b ) is done in Setting 4.1; ( c ) and ( d ) areshown in Corollaries 3.6 and 2.6. Finally, since L = rt( I ) = rn J ( I ) + 1, then I L = J I L − and f L ∈ J I L − , which defines an equation of the desired form, namely, P = P nj =1 ξ j P j + s L , with P j ∈ R [ ξ , . . . , ξ n , s ] L − . The image of the class of this equation through the isomorphism (4.2)is precisely the class of P (0 , . . . , ,
1) = 1. Note that, for d = L , then ( J I L − : f L ) = R , and theisomorphism (4.2) is given by E ( I ) L ∼ = R/ ( J I d − : f d − ), which says, in particular, that thereexists a unique top-degree equation. (cid:3) Remark 4.8.
Whenever the conditions T i,d = 0, for all d ≥ i = 1 , . . . , n , do nothold, we need to be more careful when trying to describe the equations of the Rees algebra ofthe Jacobian ideal of I . Our guide here will be Corollary 2.6. Note that we may have non-zero terms on both sides of the short exact sequence (2.6). In particular, it may happen thatrn J ( I ) + 1 < rt( I ). However, even in this case, we will have an equation of the form 4.3, with L = rn J ( I ) + 1. The rest of the equations of R ( I ) will have degree in s smaller than L , althoughthey may have total degree much bigger.5. An application to D -module theory Let X be a smooth n -dimensional complex variety and let D X be the sheaf of linear dif-ferential operators on X with holomorphic coefficients. Taking local coordinates at O ∈ X we will simply consider R = C { x , . . . , x n } and its associated ring of differential operators D R = C { x , . . . , x n }h ∂ , . . . , ∂ n i where ∂ i := ddx i are the partial derivatives with respect to the variable x i , i = 1 , . . . , n . Notice that ∂ i x i − x i ∂ i = 1 so this is a non-commutative Noetherianring whose elements can be expressed in its normal form as P := P ( x, ∂ ) = X α =( α ,...,α n ) ∈ Z n ≥ a α ( x , . . . , x n ) ∂ α · · · ∂ α n n , with finitely many a α ( x , . . . , x n ) ∈ R different from zero. The order of such a differential oper-ator is ord( P ) = max {| α | | a α = 0 } and its symbol is the element in C { x , . . . , x n } [ ξ , . . . , ξ n ] σ ( P ) = X | α | =ord( P ) a α ( x , . . . , x n ) ξ α · · · ξ α n n . Indeed, we have a filtration F := { F i } i ∈ Z ≥ of D R given by the order, or equivalently settingdeg( x i ) = 0 and deg( ∂ i ) = 1, whose associated graded ring is gr F ( D R ) ∼ = C { x , . . . , x n } [ ξ , . . . , ξ n ] , with the isomorphism given by sending P ∈ gr F ( D R ) to the symbol σ ( P ).More generally we may consider the polynomial ring D R [ s ] with coefficients in D R whoseelements are P ( s ) := P ( x, ∂, s ) = P s L + P s L − + · · · + P L , with P i ∈ D R . The total order of P ( s ) is ord T ( P ( s )) = max { ord( P i ) + i | i = 0 , . . . , L } and its total symbol is σ T ( P ( s )) = X | α | + i =ord T ( P ( s )) a α ( x , . . . , x n ) ξ α · · · ξ α n n s i ∈ C { x , . . . , x n } [ ξ , . . . , ξ n , s ] . In this case, the filtration F T := { F Ti } i ∈ Z ≥ of D R [ s ] given by the total order, or equivalentlysetting deg( x i ) = 0, deg( ∂ i ) = 1 and deg( s ) = 1, provides an isomorphism gr F T ( D R [ s ]) ∼ = C { x , . . . , x n } [ ξ , . . . , ξ n , s ] . Given f ∈ R , there exist P ( s ) ∈ D R [ s ] and a nonzero polynomial b ( s ) ∈ Q [ s ] such that P ( s ) · f f s = b ( s ) f s . The unique monic polynomial of smallest degree satisfying this functional equation is called the
Bernstein-Sato polynomial of f . This is an important invariant in the theory of singularities (see[11], for further details). The Bernstein-Sato should be understood as an equation in R f [ s ] f s ,which is the free rank-one R f [ s ]-module generated by the formal symbol f s . Moreover, R f [ s ] f s has a D R [ s ]-module structure given by the action of the partial derivatives as follows: for h ∈ R f [ s ] we have ∂ i · h f s = (cid:18) dhdx i + shf − dfdx i (cid:19) f s . Let D R [ s ] f s ⊂ R f [ s ] f s be the D R [ s ]-submodule generated by f s . This module has a presenta-tion as D R [ s ] f s ∼ = D R [ s ]Ann D R [ s ] ( f s ) , IVISORS OF EXPECTED JACOBIAN TYPE 13 where Ann D R [ s ] ( f s ) := { P ( s ) ∈ D R [ s ] | P ( s ) · f s = 0 } . The Bernstein-Sato polynomial is theminimal polynomial of the action of s on D R [ s ] f s D R [ s ] f f s ∼ = D R [ s ]Ann D R [ s ] ( f s ) + ( f ) . In order to study these annihilators we may filter them by the order of the correspondingdifferential operatorsAnn (1) D R [ s ] ( f s ) ⊆ Ann (2) D R [ s ] ( f s ) ⊆ · · · ⊆ Ann D R [ s ] ( f s ) . A lot of attention has been paid to the case that this chain stabilizes at the first step.
Definition 5.1.
A germ of divisor (
D, O ), with reduced equation given by f ∈ R , is of lineardifferential type if Ann (1) D R [ s ] ( f s ) = Ann D R [ s ] ( f s ) , that is Ann D R [ s ] ( f s ) is generated by totalorder one differential operators.Divisors of linear differential type have been considered in relation to several different problemsin D -module theory as we mentioned in the Introduction. In [7, Proposition 3.2] the authorsproved that divisors of linear Jacobian type are of linear differential type.Of course, being a divisor of linear differential type is a very restrictive condition since, ingeneral, we will have differential operators of higher total order annihilating f s . Among thesehigher order operators there exists a monic one of the form P ( s ) = s L + P s L − + · · · + P L , with ord( P i ) ≤ i , which we refer to as the Kashiwara operator (cf. [14, Theorem 6.3]). This factprompted Yano to introduce the following invariants of f (see [29], [30]). Notation 5.2.
The
Kashiwara number of f is L ( f ) := min { L | P ( s ) = s L + P s L − + · · · + P L ∈ Ann D R [ s ] ( f s ) , ord( P i ) ≤ i } . Moreover, set • r ( f ) := min { r ≥ | f r ∈ J } , • id ( f ) := min { r ≥ | f r ∈ J I r − } = integral dependence of f .Clearly, id ( f ) = rn J ( I ) + 1. Yano proved that r ( f ) ≤ id ( f ) ≤ L ( f ) . (5.1)Furthermore, if f is quasi-homogeneous we have that L ( f ) = 1.Yano [29] was able to compute the Bernstein-Sato polynomial of f , when L ( f ) = 2 , D R [ s ] ( f s ). This invariant also plays a prominent role inthe algorithm presented in [5] to compute Bernstein-Sato polynomials of isolated singularitiesthat are nondegenerate with respect to its Newton polygon.Notice that the symbol of the Kashiwara operator resembles the equation of the Rees algebraof the Jacobian ideal of top degree in s . So we would like to get deeper insight into this relation.First, since gr F T ( D R [ s ]) ∼ = C { x , . . . , x n } [ ξ , . . . , ξ n , s ], we may interpret the presentation of theRees algebra of the Jacobian ideal given in Equation 4.1 as a surjective morphism ϕ : gr F T ( D R [ s ]) −→ R ( I ) . A key result that can be found in [29, § I] (see [8, Lemma 1.9], for more details) states that σ T (Ann D R [ s ] ( f s )) ⊆ ker ϕ, where σ T (Ann D R [ s ] ( f s )) = h σ T ( P ( s )) | P ( s ) ∈ Ann D R [ s ] ( f s ) i . Indeed, it follows from [14, § σ T (Ann D R [ s ] ( f s ))) = ker ϕ. Therefore there exists some non-negative integer ℓ ∈ Z ≥ such that (ker ϕ ) ℓ ⊆ σ T (Ann D R [ s ] ( f s )).We also point out that ker ϕ is a prime ideal since R ( I ) ⊂ R [ t ] is a domain. Proposition 5.3.
Let ( D, O ) be a germ of a divisor of expected Jacobian type, with reducedequation given by f ∈ R . Let ℓ ∈ Z ≥ be the smallest non-negative integer such that (ker ϕ ) ℓ iscontained in σ T (Ann D R [ s ] ( f s )) . Then rt( I ) ≤ L ( f ) ≤ ℓ · rt( I ) . (5.2) In particular, if σ T (Ann D R [ s ] ( f s )) = ker ϕ , then rt( I ) = L ( f ) .Proof. It is readily seen that id ( f ) = rn J ( I ) + 1. Moreover, since f is of expected Jacobian type,then rt( I ) = rn J ( I ) + 1, and so, rt( I ) = id ( f ) ≤ L ( f ) (see (5.1)). Furthermore, the ideal ofequations of the Rees algebra of I contains an element of the form s r +1 + p s r + · · · + p r +1 ∈ ker ϕ, where r + 1 = rt( I ) = id ( f ), and where each p j is either zero, or else a polynomial of degree j .Therefore, we have that( s r +1 + p s r + · · · + p r +1 ) ℓ ∈ (ker ϕ ) ℓ ⊆ σ T (Ann D R [ s ] ( f s )) , so there exists a Kashiwara operator of degree at most ℓ · rt( I ). (cid:3) The upper bound in (5.2) is far from being sharp as we will see in the examples of Section 6.The issue here is how to lift an equation of the Rees algebra to a differential operator thatannihilates f s . We would like to mention that necessary conditions for the existence of such alifting were already given in [29]. 6. Examples
Let R = C { x , . . . , x n } be the ring of convergent series with coefficients in C and, for a given f ∈ m , let I = ( f , . . . , f n , f ) and J = ( f , . . . , f n ) be the Jacobian and gradient ideal of f . Weknow that J is a reduction of I . If moreover J is generated by a regular sequence, then J is oflinear type (see Setting 4.1 and Example 4.5).The aim of this section is to illustrate with some examples the condition of having the expectedrelation type and compare the relation type of I with the invariant L ( f ). In order to do so wewill use the mathematical software packages Macaulay2 [12],
Magma [4], and
Singular [10], totest the sufficient conditions T i,d = ( J i − I d − : f i ) ∩ I d − /J i − I d − = 0considered in Theorem 4.7. Of course, T ,d = 0 is always satisfied since f is a nonzero divisor.In the case of plane curves we will only have to check whether the conditions T ,d = 0 hold for d ≥ IVISORS OF EXPECTED JACOBIAN TYPE 15
Warning : Notice that we have to deal with an infinite set of conditions, namely, the vanishingof the modules T i,d . Up to now, we do not know how to solve this difficulty. In the exampleswe present we could compute T i,d for all values of d up to a positive integer much larger thanthe reduction number of I . This suggests that the examples we deal with are of the expectedJacobian type but, in no case, our computations should be considered as formal proofs. Weshould also point out that the calculations of these colon ideals tend to be extremely costly fromthe computational point of view.When computing the invariant L ( f ) we will use the Kashiwara.m2 package [3].6.1.
Some examples of plane curves.
Let f ∈ R = C { x , x } be the equation of a germof plane curve. It is proved in [20, Proposition 2.3.1] that f is a divisor of linear Jacobiantype if and only if f is quasi-homogeneous, so this is a very restrictive assumption. Indeed, forirreducible plane curves, this corresponds to the case where f = x a + y b , with gcd( a, b ) = 1,a particular case of irreducible curve with one characteristic exponent. It is well known thatthe Bernstein-Sato polynomial varies within a deformation with constant Milnor number. Weare going to test the behaviour of the expected relation type property with some examples ofirreducible plane curves with an isolated singularity at the origin. • Reiffen curves:
We consider f = x a + y b + xy b − with b ≥ a + 1, a ≥
4. This family ofirreducible plane curves with one characteristic exponent has been a recurrent example in thetheory of D -modules. In the case a = 4, Nakamura [18, 19] gave a a description of Ann D R [ s ] ( f s )and showed that L ( f ) = 2.We have tested many examples of Reiffen curves varying the values of a and b . In all the caseswe checked that T ,d = 0, for all d ≥
2. Thus, by e.g., Corollary 3.6, I has the expected relationtype with respect to J , in particular, f is a divisor of expected Jacobian type (see Definition 4.2).By Corollary 2.9 and Remark 2.10, we have E ( I ) d ∼ = ( J I d − : f d ) / ( J I d − : f d − ) and, furthercomputations suggest that ( J I d − : f d ) = (( b − x + by, y a − − d ), for all d ≤ ⌊ a/ ⌋ −
1, and(
J I d − : f d ) = R , for d ≥ ⌊ a/ ⌋ . In particular, rt( I ) = rn J ( I ) + 1 = ⌊ a/ ⌋ . • Irreducible curves with one characteristic exponent:
More generally we considerdeformations with constant Milnor number of irreducible curves with one characteristic exponentwhich have the form f = x a + y b − X t i,j x i y j , where gcd( a, b ) = 1 and the sum is taken over the monomials x i y j such that 0 ≤ i ≤ a −
2, 0 ≤ j ≤ b − bi + aj > ab . It is well known that these curves belong to the same equisingularityclass but their analytic type varies depending of the parameters t i,j . In particular, the Bernstein-Sato polynomial also varies and there exists a stratification of the space of parameters with aspecific Bernstein-Sato polynomial at each strata (see [15, 16] and [9]).We take for example the following case considered by Kato [15] · Let f = x + y − t , x y − t , x y − t , x y − t , x y . The stratification given by theBernstein-Sato polynomial with its corresponding L ( f ) invariant is: { t , = 0 , t , + 175 t , = 0 } . We have L ( f ) = 2. { t , = 0 , t , + 175 t , = 0 } . We have L ( f ) = 3. { t , = 0 , t , t , = 0 } . We have L ( f ) = 3. { t , = 0 , t , = 0 , t , = 0 } . We have L ( f ) = 2. { t , = 0 , t , = 0 , t , = 0 } . We have L ( f ) = 2. { t , = 0 , t , = 0 , t , = 0 , t , = 0 } . We have L ( f ) = 2. { t , = 0 , t , = 0 , t , = 0 , t , = 0 } . We have L ( f ) = 1.For any representative in each strata that we considered we checked out that T ,d = 0 for allthe values of d that we could compute. Moreover, the relation type is always rt( I ) = 2 exceptfor the last case which obviously corresponds to the homogeneous case. In particular, there arestrata in which we have an strict inequality rt( I ) < L ( f ).We have tested several other examples of irreducible plane curves with one characteristicexponent and all of them satisfied the conditions T ,d = 0. This suggests that this class of planecurves have the expected relation type and we can describe the module of effective relationsusing Corollary 3.6. • Irreducible curves with two characteristic exponents:
We start considering thesimplest example of such a plane curve which is: · Let f = ( y − x ) − x y . We have L ( f ) = 2. We checked out that condition T ,d = 0 issatisfied for all the values of d that we could compute and that the relation type is rt( I ) = 2 soit seems to have the expected relation type.However we can find examples with two characteristic exponents not satisfying T ,d = 0. Forexample, · Let f = ( y − x ) − x y . The reduction number of I with respect to J is rn J ( I ) = 4. Wechecked that T , = 0, T , = 0, T , = 0 and T , = 0 but T ,d = 0 for all the values d ≥ → ( f I d − : f ) ∩ I d − f · [( f I d − : f ) ∩ I d − ] + f I d − −→ E ( I ) d −→ ( J I d − : f d )( J I d − : f d − ) → . and the right term is zero for d ≥
6. Even though the conditions T ,d = 0 are not satisfied forall d ≥ d ≥
6. Thereforethe relation type is rt( I ) = 5 so f has the expected relation type. Notice that the modules ofeffective relations are not as easy to describe as in Corollary 2.9.Our computer runs out of memory before computing the invariant L ( f ).6.2. Some examples which have not the expected relation type.
Narv´aez-Macarro [20]considered some examples of non-isolated singularities which are not of linear Jacobian type.We will revisit them from our own perspective. We point out that these examples satisfy J = I so the effective relations for d ≥ L ( f ) = 1 but the relation type is strictly bigger than one. · f = xy ( x + y )( x + yz ).We have that T ,d = 0 for all the values of d that we could compute. On the other hand, T , = 0 but T ,d = 0 for all the values of d ≥ E ( I ) d ∼ = H ( f t, f t, f t ; R ( I )) d f tH ( f t, f t, f t ; R ( I )) d − , but in this case E ( I ) d = 0 for all d ≥
3, i.e. rt( I ) = 2, as it was described in [20]. IVISORS OF EXPECTED JACOBIAN TYPE 17 · f = ( xz + y )( x k − y k ).We have that T ,d = 0 for all the values of d that we could compute. However: · k = 4 : T , = 0, but T ,d = 0 for d ≥
3. This suggests that rt( I ) = 2. · k = 7 : T , = 0, T , = 0 and T , = 0 but T ,d = 0 for d ≥
5. Thus rt( I ) = 4.Here we also have E ( I ) d ∼ = H ( f t, f t, f t ; R ( I )) d f tH ( f t, f t, f t ; R ( I )) d − . For k = 4 we have H ( f t, f t, f t ; R ( I )) d = 0 if d ≥ k = 7, we have H ( f t, f t, f t ; R ( I )) d = 0 if d ≥
5. This also follows from the computations done in [20].
References [1] I. Aberbach, L. Ghezzi, H.T. H`a,
Homology multipliers and the relation type of parameter ideals , Pacific J.Math. (2006), no. 1, 1-39. 8[2] R. Arcadias,
Minimal resolutions of geometric D-modules , J. Pure Appl. Algebra (2010), 1477-1496. 1[3] G. Blanco and A. Leykin,
Kashiwara.m2 . A package for Macaulay 2 available athttps://github.com/Macaulay2/Workshop-2016-Warwick/tree/master/Dmodules. 15[4] W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. I. The user language , J. SymbolicComput., 24:235-265, 1997. 14[5] J. Brian¸con, M. Granger, Ph. Maisonobe and M. Miniconi,
Algorithme de calcul du polynˆome de Bernstein:cas non d´eg´en´er´e.
Ann. Inst. Fourier (Grenoble) (1989), 553-610. 1, 13[6] W. Bruns and H. Herzog, Cohen-Macaulay rings , Cambridge Studies in advanced mathematics , Cam-bridge University Press, Cambridge 1993. 2, 8[7] F. J. Calder´on-Moreno and L. Narv´aez-Macarro, The module D f s for locally quasi-homogeneous free divisors, Compositio Math. (2002), 59–74. 1, 2, 10, 13[8] F. J. Calder´on-Moreno and L. Narv´aez-Macarro,
On the logarithmic comparison theorem for integrable loga-rithmic connections , Proc. London Math. Soc., (2009), 585-606. 1, 10, 14[9] Pi. Cassou-Nogu`es, ´Etude du comportement du polynˆome de Bernstein lors d’une d´eformation `a µ -constantde x a + y b avec ( a, b ) = 1 , Compositio Math. (1987), 291-313. 15[10] Decker, W.; Greuel, G.-M.; Pfister, G.; Sch¨onemann, H.: Singular
Bernstein-Sato polynomials and functional equations.
Algebraic approach to differential equa-tions, 225–291, World Sci. Publ., Hackensack, NJ, 2010. 12[12] D. Grayson and M. Stillman,
Macaulay 2.
The Theory of d -Sequences and Powers of Ideals . Adv. in Mathematics (1982), 249-279. 10[14] M. Kashiwara, B-functions and holonomic systems. Rationality of roots of B-functions.
Invent. Math. (1976/77), 33-53. 1, 13, 14[15] M. Kato, The b -function of µ -constant deformation of x + y , Bull. College Sci. Univ. Ryukyus (1981),5-10. 15[16] M. Kato, The b -function of µ -constant deformation of x + y , Bull. College Sci. Univ. Ryukyus (1982),5-8. 15[17] F. Mui˜nos and F. Planas-Vilanova, The equations of Rees algebras of equimultiple ideals of deviation one,
Proc. Amer. Math. Soc. (2013), 1241-1254. 2, 3, 4, 5, 6, 8, 9[18] Y. Nakamura,
On invariants of Reiffen’s isolated singularity,
Algebraic analysis and the exact WKB analysisfor systems of differential equations, 7-13, RIMS Kˆokyˆuroku Bessatsu, B5, Res. Inst. Math. Sci. (RIMS),Kyoto, 2008. 15[19] Y. Nakamura,
The b -function of Reiffen’s ( p, isolated singularity, J. Algebra Appl. (2016), no. 6,1650115, 11 pp. 15[20] L. Narv´aez-Macarro, Linearity conditions on the Jacobian ideal and logarithmic-meromorphic comparison forfree divisors , Contemp. Math. (2008) 1, 2, 10, 15, 16, 17 [21] L. Narv´aez-Macarro,
A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors ,Advances in Mathematics (2015), 1242-1273. 1[22] D.G. Northcott and D. Rees,
Reductions of ideals in local rings , Mathematical Proceedings of the CambridgePhilosophical Society (1954), 145-158. 8[23] F. Planas-Vilanova, On the module of effective relations of a standard algebra , Math. Proc. Camb. Phil. Soc. (1998), 215-229. 2, 3[24] J.M. Ruiz,
The basic theory of Power series , Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn,Braunschweig, 1993. 10[25] I. Swanson and C. Huneke,
Integral closure of ideals, rings, and modules,
London Mathematical SocietyLecture Note Series,
Cambridge University Press, 2006. 2, 8, 9, 10[26] T. Torrelli,
Logarithmic comparison theorem and D -modules: an overview . Singularity theory, 995–1009,World Sci. Publ., Hackensack, NJ, 2007. 1[27] U. Walther, Survey on the D -module f s . With an appendix by Anton Leykin. Math. Sci. Res. Inst. Publ.,67. Vol. I, 391–430, Cambridge Univ. Press, New York, 2015. 1[28] U. Walther,
The Jacobian module, the Milnor fiber, and the D -module generated by f s , Invent. Math. (2017), 1239-1287. 1[29] T. Yano,
On the theory of b -functions, Publ. Res. Inst. Math. Sci. (1978), 111-202. 1, 13, 14[30] T. Yano, b -functions and exponents of hypersurface isolated singularities, Singularities, Part 2 (Arcata, Calif.,1981), 641-652, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983. 13
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