Multiplicity of the saturated special fiber ring of height three Gorenstein ideals

Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.


Introduction
The saturated special fiber ring is an algebra that was in introduced in [5] and that was initially motivated by the interest of studying rational maps with the use of syzygies and blow-up algebras. This algebra encodes valuable information regarding the degree and birationality of rational maps and is also related to the j-multiplicity of ideals (see [5]). Recently, for the case of perfect ideals of height two, its multiplicity was computed in [8]. In this paper, we extend the family of ideals for which the multiplicity of the saturated special fiber ring is known. Specifically, we treat the case of Gorenstein ideals of height three that are minimally generated by homogeneous polynomials of the same degree.
Let k be a field, R be the polynomial ring R = k[x 1 , . . . , x d ] and m ⊂ R be the maximal irrelevant ideal m = (x 1 , . . . , x d ). Let I ⊂ R be a homogeneous Gorenstein ideal of height three, n = µ(I) be the minimal number of generators of I, and suppose that I is minimally generated by homogeneous polynomials of the same degree δ. From the Buchsbaum-Eisenbud structure theorem [4], we can assume that I has an alternating minimal presentation matrix ϕ whose non-zero entries are all of the same degree D ≥ 1 and that I is minimally generated by the (n − 1) × (n − 1) pfaffians of ϕ. Accordingly, we can assume that I is minimally generated by n homogeneous polynomials of degree δ = 1 2 (n − 1)D. Following [5], the saturated special fiber ring of I is given by the graded k-algebra We also assume that the ideal I satisfies the condition G d , that is, µ(I p ) ≤ dim(R p ) for all p ∈ V (I) ⊂ Spec(R) such that ht(p) < d. It should be noted that the condition G d is always satisfied by generic perfect ideals of height two and by generic Gorenstein ideals of height three. We follow a strategy very similar to the one of [8], which accounts to approximate the Rees algebra with the symmetric algebra. The assumption of the condition G d allows us to the make possible that approximation. Important tools used in this paper are: the family of complexes D q • (ϕ) introduced in [27], and the computation [26,Theorem 4.4] of the multiplicity of the special fiber ring under the further assumption that I is linearly presented (i.e., when D = 1).
The following theorem contains the main result of this paper, where we compute the multiplicity of F(I).
Theorem A. Let k be a field, R be the polynomial ring R = k[x 1 , . . . , x d ] and I ⊂ R be a homogeneous ideal in R. Let n = µ(I) be the minimal number of generators of I. Suppose that the following conditions are satisfied: (iii) Every non-zero entry of an alternating minimal presentation matrix of I has degree D ≥ 1. (iv) I satisfies the condition G d .
Then, the multiplicity of the saturated special fiber ring F(I) of I is given by Next, there are some consequences of Theorem A.
The j-multiplicity of I is given by It was introduced in [1] and serves as a generalization of the Hilbert-Samuel multiplicity for non m-primary ideals. It is an interesting invariant that has encountered several applications (see [13,20,21,29,31]). The next corollary gives a formula for the jmultiplicity of any ideal satisfying the conditions of Theorem A.
Corollary B. Assume all the hypotheses and notations of Theorem A. Then, the j-multiplicity of I is given by In the second main application of Theorem A, we study rational maps that are determined by a homogeneous minimal set of generators of I. This result adds a new class of rational maps for which the syzygies of the base ideal determine the product of the degrees of the rational map and the corresponding image. The study of rational maps by using the syzygies of the base ideal is an active research topic (see e.g. [2, 5, 8-10, 12, 15, 16, 18, 22, 24, 30, 32, 33]). Let Y ⊂ P n−1 k be the closure of the image of F. Then, the following statements hold: (ii) F is birational onto its image if and only if deg P n−1 The basic outline of this paper is as follows. In Section 2, we prove Theorem A. In Section 3, we prove Corollary B and Corollary C.
2. The multiplicity of the saturated special fiber ring This section will be divided into two different subsections. In the first one, we recall a family of complexes ( [27]) that will be fundamental in our approach. In the second one, we compute the multiplicity of the saturated special fiber ring.
Throughout this paper the following setup will be used. Let n = µ(I) be the minimal number of generators of I. Assume the following: ϕ is a n × n alternating minimal presentation matrix of I.
(iv) Every non-zero entry of the matrix ϕ has degree D ≥ 1.
Remark 2.2. Note that the ideal I is minimally generated by n homogeneous polynomials all of the same degree δ := 1 2 (n − 1)D (see [4]). Remark 2.3. In terms of Fitting ideals, I satisfies the condition G d if and only if ht(Fitt i (I)) > i for all 1 ≤ i < d. So, from the presentation ϕ of I, the condition G d is equivalent to ht(I n−i (ϕ)) > i for all 1 ≤ i < d.
The following algebra is the main object of study of this paper, and we shall compute its multiplicity (under the assumptions of Setup 2.1).

Definition 2.4 ([5]
). The saturated special fiber ring of I is given by the graded kalgebra We follow an approach very similar to the one used in [8], and one of our main tools will be to use the complexes D q • (ϕ) (as introduced below in §2.1) in order to obtain approximate resolutions (see, e.g., [23], [6]) of the symmetric powers Sym q (I(δ)) of I(δ). Another important point in our proof is that, to avoid complicated Hilbert series computations, after some simple reductions we shall use the computation of [26,Theorem 4.4] (which depends upon the results of [25]).
Let A be the bigraded polynomial ring A = R[y 1 , . . . , y n ] where bideg(x i ) = (1, 0) and bideg(y i ) = (0, 1). Let S be the standard graded polynomial ring S = k[y 1 , . . . , y n ] ⊂ A. The Rees algebra can be presented as a quotient of A by using the R-homomorphism We set bideg(t) = (−δ, 1), which implies that Ψ is bihomogeneous of degree zero, and so R(I) has a structure of bigraded A-algebra. Note that we immediately obtain the following isomorphism of graded R-modules We are mostly interested in the R-grading, thus, if M is a bigraded A-module and c ∈ Z a fixed integer, then we write Notice that [M] c has a natural structure as a graded S-module. We recall the following known remark that will allow us to consider certain Hilbert functions. The special fiber ring of I is defined as F(I) := R(I)/mR(I). Since we have Following [5], to study the algebra F(I) it is enough to consider the degree zero part in the R-grading of the bigraded A-module H 1 m (R(I)). Remark 2.6. Let X be the scheme X = Proj R-gr (R(I)), where R(I) is only considered as a graded R-algebra. From [11, Theorem A4.1], we obtain the following short exact sequence [5]), we obtain the short exact sequence we have that F(I) and H 1 m (R(I)) 0 have natural structures as finitely generated F(I)-modules.
As customary, we approximate the Rees algebra with the symmetric algebra by using the following natural short exact sequence where K is the R-torsion submodule of Sym(I). The restriction to the degree zero part in the R-grading of the equality K = H 0 m (Sym(I(δ))) (Lemma 2.8(i)) and the short exact sequence (2)  2.1. The complexes of Kustin and Ulrich. In this short subsection we recall a family of complexes that was introduced in [27].
As in the proof of [26, Theorem 6.1(b)], consider the complex The above complex is interesting as the zeroth homology is Sym q (I(δ)). Here the symmetric power Sym q (I(δ)) represents the q-th graded piece of Sym(I(δ)) with respect to the S-grading. Also, under the assumptions of Setup 2.1, this complex is acyclic on the punctured spectrum of R ([26, Lemma 6.3]), that is, D q • (ϕ) p is acyclic for any p ∈ Spec(R) \ {m}.

2.2.
The computation of the multiplicity. During this subsection we compute the multiplicity of the saturated special fiber ring of the ideal I.
The following proposition deals with certain local cohomology modules and it will be an important technical tool. Proposition 2.9. Let q ≥ 1. Then, we have the following isomorphisms of graded R-modules where H d m (D q • (ϕ)) represents the complex obtained after applying the functor H d m (•) to D q • (ϕ).
Proof. The main point in the proof is that, from [26, Lemma 6.3], the complex D q • (ϕ) is acyclic on the punctured spectrum of R. Then, the result follows identically to [8,Proposition 2.7].
The next lemma contains some dimension computations, and a proof can be found in [8, Lemma 2.8].
The following proposition gives a formula which shows that, under the assumptions of Setup 2.1, the multiplicity of F(I) only depends on d, n and D. This step is fundamental in our approach since it will allow us to use the previous computation of [26,Theorem 4.4].
where β q r are the Betti numbers of (6). Therefore, the multiplicity e F(I) of F(I) only depends on the values of d, n and D. Now, the main idea is to study the homologies of the complexes H d m (D q • (ϕ)). For notational purposes we set Actually, we consider the complexes [L q • ] 0 of vector spaces over k. The additivity of Hilbert functions gives the following equalities for q ≥ 1. From Proposition 2.9 and the description (4) of the complex D q • (ϕ), we obtain that • (ϕ)) 0 = 0 for all r ≥ d + 1. Now, for all r ≤ d we define the function f r : N → N given by By using Proposition 2.9 and Remark 2.5, we have that f r corresponds with the Hilbert function of the finitely generated graded S-module H d−r m (Sym(I(δ))) 0 , that is .
Hence, for all r ≤ d, f r (q) is a polynomial for q ≫ 0 and f r (q) ∈ O q n−1 . Since (see Lemma 2.10(iii)), it follows that f r (q) ∈ O q d−2 for all r ≤ d − 2. Therefore, by summing up the above computations we obtain that Note that for q ≫ 0 we have that D q r = R(−rD) β q r (see (4)). Thus, the isomorphism yields that dim k L q r 0 = rD−1 d−1 β q r . So, for q ≫ 0, we get that The short exact sequence in (1) and Lemma 2.8(ii) give and then from the short exact sequence in (3) we have that Finally, by combining (7) and (8), for q ≫ 0 we obtain the equation So, the result follows.
The lemma below provides lower bounds for the grade of certain determinantal ideals. It will be used to obtain ideals that satisfy the conditions of Setup 2.1 and that are convenient to apply the computation of [26,Theorem 4.4]. It is likely part of the folklore, but we include a proof for the sake of completeness; for a similar setup, see [9,Proposition 4.2].
where each polynomial p i,j ∈ T is given by Proof. For any ideal J ⊂ T , denote by Rad (J) its radical ideal. Since Rad (I t−1 (M )) = Rad (I t (M )) when t is even (see, e.g., [4,Corollary 2.6]), it suffices to show that grade(I t (M )) ≥ min{m + 2 − t, d} when t is even. We proceed by induction on t and assuming that t is even with 2 ≤ t ≤ m − 1. The case t = 2 follows from [9, Lemma 4.1] since Rad(I 2 (M )) = Rad(I 1 (M )) and I 1 (M ) is generated by the p i,j 's themselves.
Therefore, depth(T Q ) ≥ min{m + 2 − t, d}, and so the result follows. Now, we are ready for the main result of this paper.
Proof of Theorem A. From Proposition 2.11, it is enough to compute the multiplicity of e F(I) for any ideal I ⊂ R that satisfies the conditions of Setup 2.1.
Without any loss of generality, assume that k is an infinite field. Let M ∈ R[z] n×n be the matrix of Lemma 2.12 with m = n. Then, by using [ such that, for any α = (α i,j,k ) ∈ U , the specialization Π α (M ) ∈ R n×n obtained by setting z i,j,k → α i,j,k satisfies the following inequalities for which the inequalities of (9) hold, and set ϕ = Π α (M ) ∈ R n×n and I = Pf n−1 (ϕ) ⊂ R. Hence, [4, Theorem 2.1, Corollary 2.6], Remark 2.3 and (9) imply that I satisfies all the conditions of Setup 2.1.

Some applications
Throughout this section, we continue using Setup 2.1.
Here we provide some applications that follow straightforwardly from the computation of Theorem A and [5]. More specifically, under the assumptions of Setup 2.1, we give an exact formula for the j-multiplicity of I and we study a rational map and so the result follows. Now, by using the formula of Theorem A, we obtain some important consequences for the rational map F in (10). So, the result is clear from Theorem A.