Divisors of expected Jacobian type

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 attention recently because of its connections with the theory of D-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some D-module theoretic invariant given by the degree of the Kashiwara operator. 1. Introduction Let (X, O) be a germ of a smooth n-dimensional complex variety and OX,O the ring of germs of holomorphic functions in a neighbourhood of O, which we identify with R=C{x1, . . . , xn}by taking local coordinates. Let DR[s] be the polynomial ring in an indeterminate swith coefficients in the ring of differential operators DR=Rh∂1, . . . , ∂niwhere ∂iare the partial derivatives with respect to the variables xi. To any hypersurface defined by f∈Rwe may attach 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 AnnDR[s](fs) := {P(s)∈DR[s]|P(s)·fs= 0},where we understand fsas a formal symbol that takes the obvious meaning frwhen specializing to any integer r∈Z. This is the defining ideal of the DR[s]-modules generated by fs, that we denote as DR[s]fs, which plays a key role in the theory of Bernstein-Sato polynomials as shown by Kashiwara [14]. Among the differential operators annihilating fsthere exists the so-called Kashiwara operator [14, Theorem 6.3] that has 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 the Kashiwara operator is an interesting analytic invariant of the singularity, although it is much coarser than the Bernstein-Sato polynomial itself. A common theme in the study of the parametric annihilator is whether it is generated by operators of degree one. This and some related linearity properties have been used by several authors in a wide range of different problems [7], [26], [20], [8] , [2], [21], [27], [28]. This linearity property of differential operators can be checked using algebraic methods as it was proved by Calder´on-Moreno and Narv´ez-Macarro in [7]. Namely, this property holds whenever the Jacobian ideal of the hypersurface fis of linear type, that is the Rees algebra and the symmetric algebra of the Jacobian ideal coincide. The aim of this paper is to get further connections between the Rees algebra of the Jacobian ideal and the parametric annihilator. Building upon work of Mui˜nos and the second author in [17], we introduce in Definition 4.2 the notion of divisors of expected Jacobian type as those divisors whose gradient ideal is of linear type and whose Jacobian ideal has relation type equal to its reduction number plus one (see also Definition 3.3). In Remark 4.3, it is easily seen that this 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 initial term or symbol of the Kashiwara operator with respect to a given order, and indeed this is the case under some extra conditions. Received by the editors September 17, 2020. Both authors are supported by the Spanish Ministerio de Econom´ıa y Competitividad MTM2015-69135-P and Generalitat de Catalunya SGR2017-932. 1 2 J. ` ALVAREZ MONTANER and F. PLANAS-VILANOVA The organization of the paper is as follows: in Section 2we review the basics on the equations of Rees algebras and recover and extend some of the results of Mui˜nos and the second author in [17]. Section 3 and 4are devoted to introduce the notion of ideal of expected relation type and its specialization to the case of the Jacobian ideal of a hypersurface. In Section 5we describe the connection between divisors of expected Jacobian type and the parametric annihilator. We relate the degree of the Kashiwara operator with the relation type of the Jacobian ideal in Proposition 5.3. In Section 6we present several examples in which we explore the case in which the ideal has the expected relation type. We also study some cases in which this condition is not 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⊆aand J⊆Iare ideals of R. 2. On the equations of Rees algebras Let (R, m) be a Noetherian local ring and let a= (f1, . . . , fm) be an ideal of R,m≥1. Let R(a) = R[at] = Ld≥0adtd⊂R[t] be the Rees algebra of a. Let A=R[ξ1, . . . , ξm] be a polynomial ring in a set of variables ξ1, . . . , ξmand coefficients in R. Consider the graded surjective morphism ϕ:A→R(a) sending ξito fit, for i= 1, . . . , n+1. The kernel of this morphism is a graded ideal Q=Ld≥1Qd, whose elements will be referred to as the equations of R(a). Let Qhdibe the ideal generated by the homogeneous equations of degree at most d. We then have an increasing sequence Qh1i ⊆ Qh2i ⊆ · · · ⊆ Qthat stabilizes at some point. The smallest integer L≥1 such that QhLi=Qis the relation type of R(a) and will be denoted rt(a). We say that ais an ideal of linear type when rt(a) = 1. Observe that the ideal Qdepends on the polynomial presentation ϕ. Nevertheless, the quotients (Q/Qhd−1i)d, for d≥2, do not (see [23]). Indeed, let α:S(a)→R(a) be the canonical graded surjective morphism between the symmetric algebra S(a) of aand the Rees algebra R(a) of a. Given d≥2, the d-th module of effective relations of ais defined to be E(a)d= ker(αd)/a·ker(αd−1). One shows that, for d≥2, E(a)d∼ =(Q/Qhd−1i)d. In particular, the relation type of acan 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∼ =H1(f1t, . . . , fmt;R(a))d, where the right-hand module stands for the degree d-component of the first Koszul homology module associated to the sequence of degree one elements f1t, . . . , fmtof R(a) ([23, Theorem 2.4]). The characterization of E(a)din terms of the Koszul homology was used in [17] in order to obtain the equations of R(a) for equimultiple ideals aof deviation one. Our purpose in this section is to rephrase, and extend a little bit, some of those results, but doing more emphasis in the Koszul conditions than in the “regular sequence type conditions”. These characterizations will be applied in the next sections to the Jacobian ideal of a hypersurface. For the sake of completeness and self-containment, we outline parts of the line of reasoning in [17]. Let us start by setting our general notations. Setting 2.1.Let (R, m) be a Noetherian local ring, n≥2. Let f1, .. . , fn∈mand f=fn+1 ∈m. Let J= (f1, . . . , fn) and I= (f1, . . . , fn, f)=(J, f) be ideals of R. For i= 1, . . . , n + 1, let Ji= (f1, . . . , fi); set J0= 0 and observe that Jn=Jand Jn+1 =I. For i= 1, . . . , n + 1 and d≥2,set Ti,d =(Ji−1Id−1:fi)∩Id−1 Ji−1Id−2. For d= 1,set Ti,1=Ji−1:fi. Note that for i= 1 (and any d≥1), then T1,d = (0 : f1)∩Id−1. For i=n+ 1 and d≥2, it was shown in [17, Proof of Lemma 3.1] that: Tn+1,d =(JId−1:f)∩Id−1 JId−2∼ =JId−1:fd JId−1:fd−1.(2.1) The isomorphism goes as follows. Given a∈(JId−1:f)∩Id−1, since Id−1=JId−1+fd−1R, write a=b+cfd−1with b∈JId−1and c∈R. The class of an element a∈(JId−1:f)∩Id−1is sent to the class of c∈JId−1:fd. DIVISORS OF EXPECTED JACOBIAN TYPE 3 Notation 2.2. A graded Koszul complex. Let us denote K(z1, . . . , zr;U) the Koszul complex of a sequence of elements z1, . . . , zrof a ring U. Since Uwill always be the Rees algebra R(I) of I, we just skip the letter U. For i= 1, . . . , n + 1, we consider the sequences fit := f1t, . . . , fitof elements of degree one in R(I); we highlight the distinct notation with the length one sequence fit. Set ft := fn+1t = f1t, . . . , fnt, ft. Thus K(fit) = K(f1t, . . . , fit;R(I)) stands for the Koszul complex associated to fit = f1t, . . . , fit, with first nonzero zero terms: K(fit) : · · · → K2(fit) ∂2 −→ K1(fit) ∂1 −→ K0(fit) →0. Let Hj(fit) = Hj(K(fit)) be its j-th homology module. Note that, since R(I) is a graded algebra, K(fit), and hence its homology, inherit a natural grading. The first nonzero terms of the degree d-component K(fit)d,d≥2, (omitting the powers of the variable t) are: · · · → K2(fit)d=∧2(Ri)⊗Id−2∂2,d−2 −→ K1(fit)d=∧1(Ri)⊗Id−1∂1,d−1 −→ K0(fit)d=Id→0. The Koszul differentials are defined as follows: if e1, . . . , eistands for the canonical basis of Riand u∈Id−2and v∈Id−1, then ∂2,d−2(ej∧el⊗u) = el⊗fju−ej⊗fluand ∂1,d−1(ej⊗v) = fjv. Note that, under the isomorphism ∧1(Ri)⊗Id−1∼ =Id−1⊕(i) · · · ⊕Id−1, the differential ∂1,d−1sends the i-th tuple (a1, . . . , ai)∈(Id−1)⊕ito the element a1f1+· · ·+aifi∈Id. In particular, for d= 1, H1(fit)1= {(a1, . . . , ai)∈Ri|Pi jajfj= 0}=Z1(f1, . . . , fi), the first module of syzygies of Ji= (f1, . . . , fi). Remark 2.3. Equations vs cycles. Let Qbe the ideal of equations of R(I). As said before, E(I)d∼ =Q Qhd−1id ∼ =H1(ft)d=H1(K(ft))d=H1(f1t, . . . , fnt, ft;R(I))d,(2.2) i.e., the d-th module of effective relations E(I)dof Iis isomorphic to the degree d-component of the first Koszul homology module H1(ft) of ft, where d≥2. This isomorphism sends the class of an equation P∈Qdto the class of the cycle (P1(f), . . . , Pn(f), Pn+1(f)) ∈Ln+1 j=1 Id−1, where f = f1, . . . , fn, f, P=Pn+1 j=1 ξjPj, and Pj∈Ad−1=R[ξ1, . . . , ξn+1]d−1. (See [17, Remark 2.1].) The next two remarks are devoted to write more explicitly some complexes and morphisms that will be used subsequently. Remark 2.4. A short exact sequence of Koszul complexes. Let K(fit) be the Koszul complex associated to the length one sequence fit∈R(I). So K0(fit) = R(I), K1(fit) = ∧1(R)⊗R(I)∼ =R(I), and Kj(fit) = 0, for j6= 0,1. In degree d≥1, K0(fit)d=Id,K1(fit)=(∧1(R)⊗R(I))d∼ =Id−1and, for a∈Id−1, then ∂1,d−1(a) = afi. There is an isomorphism of Koszul complexes K(fit) ∼ =K(fi−1t) ⊗K(fit). Concretely, Kp(fit) ∼ =M r+s=p Kr(fi−1t) ⊗Ks(fit) = Kp(fi−1t) ⊗K0(fit)⊕Kp−1(fi−1t) ⊗K1(fit)∼ = Kp(fi−1t) ⊗R(I)⊕Kp−1(fi−1t) ⊗R(I)∼ =Kp(fi−1t) ⊕Kp−1(fi−1t), which induces a short exact sequence of Koszul complexes: 0→K(fi−1t) →K(fit) →K(fi−1t)(−1) →0,(2.3) where K(fi−1t)(−1) is the shifted complex by -1, i.e., Ks(fi−1t)(−1) = Ks−1(fi−1t). In particular, for d≥1, the degree d-component gives rise to the the short exact sequence of complexes: 0→K(fi−1t)d→K(fit)d→K(fi−1t)(−1)d→0. 4 J. ` ALVAREZ MONTANER and F. PLANAS-VILANOVA Displaying by columns the first nonzero terms of each complex, we get: . . .  . . .  . . .  0//∧2(Ri−1)⊗Id−2//  ∧2(Ri)⊗Id−2//  Ri−1⊗Id−2//  0 0//Ri−1⊗Id−1//  Ri⊗Id−1//  Id−1//  0 0//Id//  Id//  0//  0 0 0 0 . The middle row, 0 →K1(fi−1t)d→K1(fit) →K1(fi−1t)(−1)d→0, is nothing else than: 0→Id−1⊕(i−1) · · · ⊕Id−1−→ Id−1⊕(i) · · · ⊕Id−1→Id−1→0, where the first morphism sends (a1, . . . , ai−1) to (a1, . . . , ai−1,0), the inclusion, and the second morphism sends (a1, . . . , ai) to ai, 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≥1. · · · → H2(K(fi−1t)(−1))d δ −→ H1(fi−1t)d→H1(fit)d→H1(K(fi−1t)(−1))d δ −→ →H0(fi−1t)d→H0(fit) →H0(K(fi−1t)(−1))d→0. Clearly, Hj(K(fi−1t)(−1))d=Hj−1(fi−1t)d−1and H0(K(fi−1t)(−1))d= 0. The connecting morphism is known to be the multiplication by the element ±fit. Thus we get: · · · → H1(fi−1t)d−1 ·±fit −→ H1(fi−1t)d→H1(fit)d→H0(fi−1t)d−1 ·±fit −→ H0(fi−1t)d→H0(fit)d→0. If d= 1, then H1(fi−1t)0= 0, H0(fi−1t)0=Rand H0(fi−1t)1=I/Ji−1. Hence ker H0(fi−1t)0 ·±fit −→ H0(fi−1t)1= (Ji−1:fi) = Ti,1. In particular, for i= 1, . . . , n + 1 and d= 1, one deduces the exact sequence: 0→H1(fi−1t)1→H1(fit)1→Ti,1→0,(2.4) where H1(fi−1t)1=Z1(f1, . . . , fi−1) and H1(fit)1=Z1(f1, . . . , fi). If d≥2, one can check that H0(fi−1t)d−1=Id−1/Ji−1Id−2and H0(fi−1t)d=Id/Ji−1Id−1. Thus ker H0(fi−1t)d−1 ·±fit −→ H0(fi−1t)d= (Ji−1Id−1:fi)∩Id−1/Ji−1Id−2=Ti,d. (See Setting 2.1.) In particular, for i= 1, . . . , n + 1 and d≥2, we deduce the exact sequence: H1(fi−1t)d−1 ·±fit −→ H1(fi−1t)d→H1(fit)d→Ti,d →0.(2.5) Note that the middle morphism in (2.5) is induced by the inclusion. Namely, the class of a cycle (a1, . . . , ai−1), aj∈Id−1, maps to the class of the cycle (a1, . . . , ai−1,0). Similarly, the right-hand morphism is induced by the projection (a1, . . . , ai)7→ ai. DIVISORS OF EXPECTED JACOBIAN TYPE 5 We recover [17, Lemma 3.1]. Keeping the notations as in Setting 2.1 and Notation 2.2: Corollary 2.6.For d≥2, the following sequence is exact. 0→H1(fnt)d ft ·H1(fnt)d−1 −→ E(I)d−→ JId−1:fd JId−1:fd−1→0.(2.6) The right-hand morphism sends the class of an equation P∈Qdto the class of P(0,...,0,1). Proof. Take i=n+ 1 and d≥2 in (2.5). Then H1(fn+1t)d=H1(ft)d, which by (2.2), is isomorphic to E(I)d. See also the definition of Tn+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∈Qdis sent to the class of (P1(f), . . . , Pn+1(f)) ∈Ln+1 j=1 Id−1through (2.2), where P=Pn+1 j=1 ξjPj,Pj∈Ad−1. By (2.5), (P1(f), . . . , Pn+1(f)) is sent to the class of Pn+1(f) ∈(JId−1:f)∩Id−1. Write Pn+1 =Pn j=1 ξjQj+cξd−1 n+1, with Qj∈Ad−2and c∈R. In particular, Pn+1(f) = b+cfd−1, with b=Pn j=1 fjQj(f) ∈JId−2. Then the isomorphism (2.1) sends the class of Pn+1(f) to the class of c∈JId−1:fd. Observe that P(0,...,0,1) = Pn+1(0,...,0,1) = c. The first part of the following result is shown in [17, Lema 3.3]. Our proof here is a direct consequence of Remarks 2.4 and 2.5, and the sequences (2.4) and (2.5). We keep the notations as in Setting 2.1 and Notation 2.2. Theorem 2.7.Fix d≥1and i= 1, . . . , n + 1. (a)The following two conditions are equivalent: (i)H1(f1t)d= 0, H1(f2t)d= 0, . . . , H1(fit)d= 0; (ii)T1,d = 0, T2,d = 0, . . . , Ti,d = 0. (b)Suppose that, for some i= 1, . . . , n,T1,d = 0, T2,d = 0, . . . , Ti,d = 0. Then H1(fi+1t)d∼ =Ti+1,d. (c)Fix now d≥2. Suppose that, for some i= 1, . . . , n −1, T1,d = 0, T2,d = 0, . . . , Ti,d = 0 and that T1,d−1= 0, T2,d−1= 0, . . . , Ti,d−1= 0. Then the following sequence is exact. 0→(JiId−1:fi+1)∩Id−1 fi+2 ·[(JiId−2:fi+1)∩Id−2] + JiId−2→H1(fi+2t)d→Ti+2,d →0.(2.7) Proof. The implication (i)⇒(ii) follows directly from the exact sequences (2.4) and (2.5). Since H1(f0t) = 0, then, by (2.4) and (2.5), H1(f1t)d∼ =T1,d and (ii)⇒(i) holds for i= 1. Suppose that (ii)⇒(i) holds for i−1≥1. By the induction hypothesis, H1(fi−1t)d= 0 and, by (2.4) and (2.5), H1(fit)d∼ =Ti,d = 0. This proves (a). Suppose now that, for some i= 1, . . . , n −1, T1,d = 0, T2,d = 0, . . . , Ti,d = 0. In particular, since (ii)⇒(i), H1(f1t)d= 0, H1(f2t)d= 0, . . . , H1(fit)d= 0. Using (2.4) and (2.5), for the integer i+ 1, H1(fi+1t)d∼ =Ti+1,d. This proves (b). Suppose now that the hypotheses in (c) hold. Then, by (b) applied to d−1, we obtain the isomorphism H1(fi+1t)d−1∼ =Ti+1,d−1. Therefore, H1(fi+1t)d−1∼ =(JiId−2:fi+1)∩Id−2 JiId−3and H1(fi+1t)d∼ =(JiId−1:fi+1)∩Id−1 JiId−2. Through these isomorphisms, H1(fi+1t)d fi+2t·H1(fi+1t)d−1 ∼ =(JiId−1:fi+1)∩Id−1 fi+2 ·[(JiId−2:fi+1)∩Id−2] + JiId−2. The rest follows from the exact sequence (2.5) applied to i+ 2. 6 J. ` ALVAREZ MONTANER and F. PLANAS-VILANOVA Next we specialise Theorem 2.7 (c), to the case n= 2. Corollary 2.8.Let (R, m)be a Noetherian local ring, f1, f2, f ∈mand let J= (f1, f2)and I= (f1, f2, f). Fix d≥2. Assume that (0 : f1)∩Id−2= 0 and (0 : f1)∩Id−1= 0. Then the following sequence is exact. 0→(f1Id−1:f2)∩Id−1 f·[(f1Id−2:f2)∩Id−2] + f1Id−2−→ E(I)d−→ (JId−1:fd) (JId−2:fd−1)→0.(2.8) Proof. Take n= 2, i= 1 and d≥2 in Theorem 2.7 (c). Then T1,d−1= (0 : f1)∩Id−2and T1,d = (0 : f1)∩Id−1, which are zero by hypothesis. The rest follows from the sequence (2.7). Corollary 2.9.Let (R, m)be a Noetherian local ring, f1, f2, f ∈mand let J= (f1, f2)and I= (f1, f2, f). Fix L≥2. Suppose that (0 : f1)∩Id−1= 0, for all 1≤d≤L, and that (f1:f2)⊆(f1:f). Then the following two conditions are equivalent. (a)T2,d = (f1Id−1:f2)∩Id−1/f1Id−2= 0, for all 2≤d≤L; (b)E(I)d∼ =(JId−1:f2)/(JId−2:fd−1), for all 2≤d≤L. Proof. The hypotheses (0 : f1)∩Id−1= 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 T2,d ensures the vanishing of the left-hand side term in the exact sequence (2.8). Thus (b) holds. Conversely, assume that (b) holds. Let us prove T2,d = 0, by induction on d, 2 ≤d≤L. So take d= 2. Using (b) and (2.8), for d= 2, then (f1I:f2)∩I=f·(f1:f2). Using the hypothesis (f1:f2)⊆(f1:f), we get (f1I:f2)∩I=f·(f1:f2)⊆f·(f1:f)⊆(f1). Thus T2,2= 0. Take now d≥3, d≤L. By the induction hypothesis T2,d−1= 0, so (f1Id−2:f2)∩Id−2=f1Id−3. Using (b) and (2.8), for such d, then (f1Id−1:f2)∩Id−1=f·[(f1Id−2:f2)∩Id−2] + f1Id−2=f·[f1Id−3∩Id−2] + f1Id−2⊆f1Id−2. Hence T2,d = 0. Remark 2.10.In the case that f1, f2is a regular sequence, then clearly (0 : f1)∩Id−1= 0 for all 1≤d≤Land (f1:f2) = f1R⊆(f1:f). However the converse does not always hold as the next example shows. Example 2.11.Let R=k[[x, y]] be the formal power series ring in two variables over a field kof characteristic zero. Take a, b ≥2 and consider the ideals J= (f1, f2) and I= (f1, f2, f) with f=xayb, f1=df dx =axa−1yband f2=df dy =bxayb−1. Then (f1:f2) = yR ⊆(f1:f) = R, whereas f1, f2is not a regular sequence. We point out that J=Iis an ideal of linear type. 3. 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 aand bbe two ideals of R. The ideal bis areduction of aif b⊆aand there is an integer r≥0 such that ar+1 =bar. From the definition 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 ais always a reduction of itself. An ideal awhich has no reduction other than itself is called a basic ideal. The smallest integer r≥0 satisfying the equality equality ar+1 =baris called the reduction number of awith respect to band is denoted rnb(a). For b=a, rnb(a) = 0. (see [22]). DIVISORS OF EXPECTED JACOBIAN TYPE 7 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≥2, and let J= (f1, . . . , fn)be a reduction of I= (f1, . . . , fn, f). Then rnJ(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), Tn+1,d = 0, for all d≥n+ 1. Therefore (JId−1:fd)=(JId−2:fd−1), for all d≥L+ 1. Since Jis a reduction of I, then (JIm−1:fm) = R, for m0 large enough. Thus fL∈JIL−1,IL=JIL−1and rnJ(I)≤L−1. Definition 3.3.Let (R, m) be a Noetherian local ring and let J= (f1, . . . , fn) be a reduction of I= (f1, . . . , fn, f). We say that Ihas the expected relation type with respect to Jif rnJ(I) + 1 = rt(I). When Jis 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/man infinite field. Let abe an ideal of R. (a) If ais of linear type, then ais basic and has the expected relation type. (b) If ais a parametric ideal, that is, generated by a system of parameters, then ais basic, but it is not necessarily of linear type, nor it has necessarily the expected relation type. Proof. Let bbe a reduction of a. By [22, 2. Theorem 1], there exists an ideal c⊆b⊆a, which is a minimal reduction of a, that is, no ideal strictly contained in cis a reduction of a. By [22, 2. Lemma 3], every minimal set of generators of c= (x1, . . . , xs) can be extended to a minimal set of generators of a= (x1, . . . , xs, xs+1, . . . , xm), with µ(c) = s≤µ(a) = m, where µ(·) stands for the minimal number of generators. If ais of linear type, then S(a)∼ =R(a). On tensoring by A/m,k[T1, . . . , Tm]∼ =F(a), where F(a) = ⊕d≥0ad/madis the fiber cone of a. On taking Krull dimensions, 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=aand ais basic. In particular, rnb(a) = 0 and, since ais of linear type, rt(a) = 1 = rnb(a) + 1. This proves (a). If ais a parametric ideal, then height (a) = µ(a). By [22, 4. Theorem 5], ais basic. Take now R=k[[x, y, z, w]], where w2=wz = 0, and a= (xm−1y+zm, xm, ym), m≥2. Then ais a parameter ideal, hence a basic ideal, but its relation type is at least m(see [1, Example 2.1]). The following result gives a characterization of ideals with expected relation type in terms of the Koszul homology. Proposition 3.5.Let (R, m)be a Noetherian local ring and let J= (f1, . . . , fn)be a reduction of I= (f1, . . . , fn, f). The following conditions are equivalent. (a)Ihas the expected relation type; (b)H1(fnt)d=ft ·H1(fnt)d−1, for all d≥rnJ(I)+2. In particular, if Ti,d = (Ji−1Id−1:fi)∩Id−1/Ji−1Id−1= 0, for all d≥rnJ(I)+2 and all i= 1, . . . , n, then Ihas the expected relation type. Proof. Set r= rnJ(I). If Ihas the expected relation type, then E(I)d= 0, for all d≥r+ 2. In particular, using the exact sequence (2.6), we deduce H1(fnt)d=ft ·H1(fnt)d−1, for all d≥r+ 2. Conversely, if H1(fnt)d=ft ·H1(fnt)d−1, for all d≥r+ 2, then by (2.6), E(I)d∼ =(JId−1:fd)/(JId−1: fd−1). However, if d≥r+ 2, then fd−1∈JId−2, so (JId−1:fd−1) = Rand E(I)d= 0. Therefore, rt(I)≤r+1 = rnJ(I)+1. The other inequality follows from Proposition 3.2. This shows the equivalence (a)⇔(b). If T1,d =, T2,d = 0, . . . , Tn,d = 0, for all d≥rnJ(I) + 2, by Theorem 2.7,H1(fnt)d= 0, for all d≥rnJ(I) + 2, and, by (b)⇒(a), Ihas the expected relation type. 8 J. ` ALVAREZ MONTANER and F. PLANAS-VILANOVA The main result of Mui˜nos and the second author in [17], gives an instance of ideals of expected relation type. We rephrased it here in terms of our Ti,d. Corollary 3.6.Let (R, m)be a Noetherian local ring, n≥2. Let f1, . . . , fn∈mand f∈m. Let J= (f1, . . . , fn)and I= (f1, . . . , fn, f)be ideals of R. Assume that for all d≥2and all i= 1, . . . , n, Ti,d = (Ji−1Id−1:fi)∩Id−1/Ji−1Id−2= 0. Then, for all d≥2, E(I)d∼ =(JId−1:fd) (JId−2:fd−1). In particular, if Jis a reduction of I, then Ihas the expected relation type with respect to J. Proof. If T1,d = 0, T2,d = 0, . . . , Tn,d = 0, for all d≥2, then, by Theorem 2.7 (a), H1(fnt)d= 0, for all d≥2. By the exact sequence (2.6), E(I)d∼ =(JId−1:fd)/(JId−2:fd−1). The second assertion follows directly from Proposition 3.5. 4. Divisors of expected Jacobian type Let (X, O) be a germ of a smooth n-dimensional complex variety and OX,O the ring of germs of holomorphic functions in a neighbourhood of O, which we identify with R=C{x1, . . . , xn}by taking local coordinates. Let (D, O) be a germ of divisor defined locally by f∈Rand set fi=df dxifor i= 1 ...,n. From now on, until the end of the paper, we consider the following notations. Setting 4.1.Let R=C{x1, . . . , xn}be the convergent power series ring, which is a Noetherian regular local ring (see, e.g., [25, Lemma 7.1]). Let f∈m. Set fi=df dxi, for i= 1 ...,n. Let J= (f1, . . . , fn) and I= (f1, . . . , fn, f) = (J, f). Note that if f∈m2, then fi∈mand J⊆I⊆m. The ideals Jand Iare called the gradient ideal of fand the Jacobian ideal of f, respectively. It is known that, when f∈m, then f∈(x1f1, . . . , xnfn)⊆J, where Hstands for the integral closure of the ideal H(see [25, Corollary 7.1.4]). In particular, Jis 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 Jacobian type if Iis an ideal of linear type ([8, Definition 1.11]); fwill be said of expected Jacobian type, if Jis of linear type and Ihas 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 fis of linear Jacobian type, then Iis basic, thus J=Iis of linear type and Ihas the expected relation type since rt(I) = 1 and rnJ(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 f6∈ C[λx +y],C[x+µy], for all λ, µ ∈C. Then Jis an ideal of linear type minimally generated by two elements. Proof. Since Ris local and J= (f1, f2), to see that Jis minimally generated by two elements it is enough to prove that f16∈ (f2) or f26∈ (f1). Let us see that if f1∈(f2), then fis either in C[y], or else in C[x+µy], for some µ∈C(similarly, one would do the same if f2∈(f1)). Since f6∈ C[y], degx(f) = r≥1. Write f=Pr i=0 xrgi(y) and suppose that f1=pf2, for some element p∈C{x, y}. Since f1and f2are polynomials, then p∈C[x, y] must be a polynomial too. Equating the highest degree terms in xin the expression f1=pf2, one deduces that either p= 0, or else g0 r(y) = 0. However, if p= 0, then f1= 0 and f∈C[y], a contradiction. Thus g0 r(y) = 0 and gr(y) = ar∈C,ar6= 0, because degx(f) = r≥1. Substituting gr(y) = arin fand equating again the highest degree term in xin the equality f1=pf2, one gets pg0 r−1(y) = rar6= 0. Thus p∈C,p6= 0. Setting µ= 1/p, we have gr−1(y) = rarµy +ar−1. Again, substituting this expression in fand equating the r−2 degree terms in DIVISORS OF EXPECTED JACOBIAN TYPE 9 the equality f1= (1/µ)f2, one gets gr−2(y) = arr 2(µy)2+ar−1µy +ar−2. Proceeding recursively, one would get the equality f=Pr i=0 ai(x+µy)iand so fwould be an element of C[x+µy], a contradiction. Therefore, Jis minimally generated by two elements. Now apply [13, Proposition 1.5]. Thus Jcan be generated by two elements which form a d-sequence. In particular Jis of linear type (see, e.g., [25, Corollary 5.5.5]). Example 4.5.Suppose that J= (f1, . . . , fn) is generated by an R-regular sequence, for instance, if fhas an isolated singularity at O(see, e.g., [24, IV, Remark 2.5]). In particular, Jis of linear type ([25, Corollary 5.5.5]). There are three possibilities according to the previous definition. If Iis of linear type, then fis a divisor of linear Jacobian type. Suppose that Iis not of linear type. Recall that, by the argument in Setting 4.1,Jis a reduction of Iand, by Proposition 3.2, rnJ(I)+1≤rt(I). If the equality holds, then fis of expected Jacobian type. The third and last case occurs when Jis of linear type, but rnJ(I)+1<rt(I), i.e., fis not of expected 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 linear Jacobian type are Euler homogeneous. Recall that a divisor Dis Euler homogeneous if there is a vector field χat Osuch that χ(f) = f, or in other words, f∈J. In particular, J=Iand rnJ(I) = 0. For divisors satisfying the conditions Ti,d = 0 we can explicitly describe the equations of the Rees algebra of the Jacobian ideal which corresponds to describe the blow-up at the singular locus of the divisor. More precisely, and summarizing several results in a unique statement: Theorem 4.7.Let R=C{x1, . . . , xn}be the convergent power series ring Let f∈m2,Jand Ibe as in Setting 4.1. Suppose that Ti,d = 0, for all d≥1and all i= 1, . . . , n. Set ξi+1 =s. Let ϕ:R[ξ1, . . . , xn, s] = C{x1, . . . , xn}[ξ1, . . . , xn, s]→R(I) (4.1) be a polynomial presentation of R(I), sending ξito fitand sto ft. Let Q=Ld≥1Qdthe ideal of equations of R(I). The following conditions hold. (a)f1, . . . , fnis an R-regular sequence and Jis of linear type. (b)Jis a reduction of Iand rt(I) = rnJ(I)+1. If f∈J, then fis a divisor of linear Jacobian type; otherwise, fis a divisor of expected Jacobian type. (c)For all d≥2, E(I)d∼ =(Q/Qhd−1i)d∼ =(JId−1:fd) (JId−2:fd−1); (4.2) the class of P(ξ1, . . . , ξn, s)∈Qdis sent to the class of P(0,...,0,1) ∈(JId−1:fd). (d)Set L= rt(I). A minimal generating set of equations of R(I)can be obtanied from a minimal generating set of Q1, the first syzygies of I, and representatives of inverse images of a minimal generating set of (JId−1:fd)/(JId−2:fd−1), for all 2≤d≤L. (e)There exists a unique top-degree equation of degree L, which is of the form sL+p1sL−1+· · · +pL,(4.3) where pj∈R[ξ1, . . . , ξn]are either zero, or else, polynomials of degree j. Proof. Since f∈m2, then fi∈mand so J⊆I⊆m. Recall that Ti,1= (Ji−1:fi). Therefore, Ti,1= 0, for all i= 1, . . . , n is equivalent to f1, . . . , fnbeing an R-regular sequence. In particular, Iis of linear type. This proves (a); (b) is done in Setting 4.1; (c) and (d) are shown in Corollaries 3.6 and 2.6. Finally, since L= rt(I) = rnJ(I) + 1, then IL=JIL−1and fL∈JIL−1, which defines an equation of the desired form, namely, P=Pn j=1 ξjPj+sL, with Pj∈R[ξ1, . . . , ξn, s]L−1. The image of the class of this equation through the isomorphism (4.2) is precisely the class of P(0,...,0,1) = 1. Note that, for d=L, then (JIL−1:fL) = R, and the isomorphism (4.2) is given by E(I)L∼ =R/(JId−2:fd−1), which says, in particular, that there exists a unique top-degree equation. 16 J. ` ALVAREZ MONTANER and F. PLANAS-VILANOVA Departament de Matem` atiques Universitat Polit` ecnica de Catalunya. Diagonal 647, Barcelona, Catalonia E-mail: [email protected] E-mail: [email protected]