THE LEADING IDEAL OF A COMPLETE INTERSECTION OF HEIGHT TWO IN A 2-DIMENSIONAL REGULAR LOCAL RING

SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

Abstract. Let (S, n) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I^{∗} be the leading ideal of I in the associated graded ring gr_{𝔫}(S), and set R = S/I and 𝔪 = 𝔫/I. In [GHK2], we prove that if μ_{G}(I^{∗}) = n, then I^{∗} contains a homogeneous system {ξ_{i}}_{1≤i≤n} of generators such that deg ξ_{i} + 2 ≤ deg ξ_{i+1} for 2 ≤ i ≤ n−1, and ht_{G}(ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = 1, and we describe precisely the Hilbert series H(gr_{𝔪}(R), λ) in terms of the degrees c_{i} of the ξ_{i} and the integers d_{i}, where d_{i} is the degree of D_{i} = GCD(ξ_{1}, …, ξ_{i}). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c_{1} + 1, an ascending sequence of positive integers (c_{1}, c_{2}, …, c_{n}), and a descending sequence of integers (d_{1} = c_{1}, d_{2}, …, d_{n} = 0) such that c_{i+1} − c_{i} > d_{i−1} − d_{i} > 0 for each i with 2 ≤ i ≤ n − 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.

1. Introduction

This paper examines generators of the leading ideal of a complete intersection of

height two in a 2-dimensional regular local ring. Motivation for our work comes from

a paper of S. C. Kothari [K] that answers several questions raised by Abhyankar

concerning the local Hilbert function of a pair of plane curves. Before going ahead,

let us fix some notation, which we shall maintain throughout this paper.

Setting 1.1. Let (S, 𝔫) be a regular local ring of dimension 2 and let I = (f, g)

be an ideal in S generated by a regular sequence f, g of length two. For simplicity

Date: May 15, 2007. 1991 Mathematics Subject Classification. Primary: 13A30, 13C05; Secondary: 13E05, 13H15. Key words and phrases. ideal of initial forms, associated graded ring, multiplicity, regular local

ring, Hilbert series, minimal generating set .

Shiro Goto, Department of Mathematics, School of Science and Technology Meiji University, 214-8571 Japan, email: goto@math.meiji.ac.jp, Shiro Goto is supported by the Grant-in-Aid for Scientific Researches in Japan (C(2), No.1364044).

William Heinzer, Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, email: heinzer@math.purdue.edu.

Mee-Kyoung Kim, Department of Mathematics, Sungkyunkwan University Jangangu Suwon 440-746, Korea, email: mkkim@skku.edu, Mee-Kyoung Kim was supported by Faculty Research Fund, Sungkyunkwan University, 2003.

2 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

we assume that the residue class field k = S/ 𝔫 is infinite. We put R = S/I and 𝔪 = 𝔫 /I. Let

R′(𝔫) = ∑_{i∈ℤ}𝔫^{i} t^{i} ⊆ S[t, t^{−1}] and R′(𝔪) = ∑_{i∈ℤ}𝔪^{i}t^{i} ⊆ R[t, t^{−1}]

denote the Rees algebras of 𝔫 and 𝔪 respectively, where t is an indeterminate. We put

G = gr_{𝔫}(S) = R′(𝔫)/t^{−1}R′(𝔫) and gr_{𝔪}(R) = R′(𝔪)/t^{−1}R′(𝔪).

For each 0 ≠ h ∈ S let o(h) = sup\{i ∈ ℤ | h ∈ 𝔫^{i}\} and put h^{∗} = \overline{ht^{n}}, where n = o(h) and \overline{ht^{n}} denotes the image of ht^{n} in G. The canonical map S → R induces the epimorphism φ : G → gr_{𝔪}(R) of the associated graded rings. We put

I^{∗} = Ker (G →^{φ} gr_{𝔪}(R)).

Then the homogeneous components {[I^{∗}]_{i}}_{i∈ℤ} of the leading form ideal I^{∗} of I are given by

[I^{∗}]_{i} = \{\overline{ht^{i}} | h ∈ I ∩ 𝔫^{i}\}

for each i ∈ ℤ. We throughout assume that a = o(f) ≤ b = o(g) and that f^{∗} ∤ g^{∗} in G. The latter part of the condition is equivalent to saying that f^{∗}, g^{∗} form a part of a minimal homogeneous system of generators of I^{∗}.

Let ℓ_{S}(∗) denote length over S. Kothari in [K] proves that

0 ≤ dim_{k}[gr_{𝔪}(R)]_{i} − dim_{k}[gr_{𝔪}(R)]_{i+1} ≤ 1

for all i ≥ a and that ℓ_{S}(R) ≥ ab; moreover, one has the equality ℓ_{S}(R) = ab if and only if f^{∗}, g^{∗} are coprime in G, that is, f^{∗}, g^{∗} form a G-regular sequence.

F. Macaulay in a 1904 paper [M] employs a different method to determine the same necessary condition as Kothari on the Hilbert function of a pair of plane curves. Using his inverse systems, Macaulay establishes the structure of the Hilbert function H(A) of a complete intersection quotient A = k[[x, y]]/(f, g) to be of the form

H = (1, 2, …, a, t_{a}, …, t_{j}, 0), (1)

where a ≥ t_{a} ≥ t_{a+1} ≥ ⋯ ≥ t_{j} = 1 and |t_{i} − t_{i+1}| ≤ 1 for all i. Thus the Hilbert function H after an initial rising segment breaks up into platforms and regular flights of descending stairs, each step of height one. The structure of H(A) is studied from the point of view of parametrizations by J. Briançon [Br] and by A. Iarrobino [Ia1] and [Ia2]. These authors prove that every sequence satisfying the conditions in

THE LEADING IDEAL OF A COMPLETE INTERSECTION 3

Equation 1 is realizable as the Hilbert function H(A) of some Gorenstein Artin algebra of the form A = k[[x, y]]/(f, g).

Let v(H) = 2 + #{platforms}. Iarrobino [Ia1], [Ia2] proves that I^{∗} needs two initial generators f^{∗}, g^{∗} and requires a new generator following each platform, and that v(H) is the minimum possible number of generators of a graded ideal defining a standard algebra with Hilbert function H. In [Ia1, Theorem 2.2.A], Iarrobino characterizes those graded ideals corresponding to I^{∗} for which I is a complete intersection of height two. He proves they are exactly the graded ideals with v(H) generators. In the paper [GHK2] we prove the following.

Theorem 1.2. [GHK2, Theorem 1.2] Let notation be as in Setting 1.1 and assume that n = μ_{G}(I^{∗}). Then I^{∗} contains a homogeneous system {ξ_{i}}_{1≤i≤n} of generators that satisfy the following three conditions.

(1) ξ_{1} = f^{∗} and ξ_{2} = g^{∗}.

(2) degξ_{i} + 2 ≤ degξ_{i+1} for each i with 2 ≤ i ≤ n − 1.

(3) ht_{G}(ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = 1.

Let {ξ_{i}}_{1≤i≤n} be a homogeneous system of generators of I^{∗} satisfying conditions (1) and (2) in Theorem 1.2. Then the ideals {(ξ_{j} | 1 ≤ j ≤ i)G}_{1≤i≤n} are independent of the particular choice of the family {ξ_{i}}_{1≤i≤n} and are uniquely determined by I. We put D_{i} = GCD(ξ_{j} | 1 ≤ j ≤ i) and d_{i} = degD_{i}. We then have the descending sequence of integers

a = d_{1} > d_{2} > ⋯ > d_{n−1} > d_{n} = 0,

and we also have \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) for all 1 ≤ i ≤ n − 1 [GHK2, Lemma 3.2]. Let c_{i} = deg ξ_{i} and let H(gr_{𝔪}(R), λ) = ∑_{i=0}^{∞} dim_{k}[gr_{𝔪}(R)]_{i}λ^{i} denote the Hilbert series of gr_{𝔪}(R). Theorem 1.3 explicitly describes H(gr_{𝔪}(R), λ) and the difference ℓ_{S}(R) − ab in terms of the integers c_{i} and d_{i}, thus sharpening Kothari’s results.

Theorem 1.3. [GHK2, Theorem 1.3] Let notation be as in Setting 1.1 and assume that n = μ_{G}(I^{∗}). The following assertions hold true.

(1) H(gr_{𝔪}(R), λ) = ^{\frac{∑_{i=2}^{n} λ^{d_{i}}(1−λ^{d_{i−1}−d_{i}})(1−λ^{c_{i}−d_{i}})}{(1−λ)^{2}}}.

(2) ℓ_{S}(R) = ∑_{i=2}^{n}(d_{i−1} − d_{i})(c_{i} − d_{i}) = ab + ∑_{i=2}^{n−1} d_{i}·[(c_{i+1} − c_{i}) − (d_{i−1} − d_{i})].

(3) c_{i+1} − c_{i} > d_{i−1} − d_{i} > 0 for each i with 2 ≤ i ≤ n − 1.

(4) [K, Corollary1] ℓ_{S}(R) = ab if and only if n = 2, i.e., f^{∗}, g^{∗} is a G-regular sequence.

In Section 2, we use the inverse of a transformation considered by Kothari and described in parts (1) and (2) of [GHK2, Corollary 2.5] to establish the existence of examples showing that every Hilbert series described in Theorem 1.3 is realizable as the Hilbert series of gr_{𝔪}(R) for some complete intersection ideal I = (f, g). Thus the conditions given in Theorem 1.3 are both necessary and sufficient for there to exist a complete intersection ideal I = (f, g) with Hilbert series

H(gr_{𝔪}(R), λ) = \frac{∑_{i=2}^{n} λ^{d_{i}}(1 − λ^{d_{i−1}−d_{i}})(1 − λ^{c_{i}−d_{i}})}{(1 − λ)^{2}}.

Remark 1.4. In the case where f and g are a regular sequence in a regular local ring S with dim S > 2, it is still true that ht_{G}(f^{∗}, g^{∗}) > 1 implies f^{∗}, g^{∗} is a G-regular sequence, and therefore I^{∗} = (f^{∗}, g^{∗})G also in this case. Thus if ht_{G}(f^{∗}, g^{∗}) = 1 and if we set D_{2} = GCD(f^{∗}, g^{∗}) and d_{2} = degD_{2}, then f^{∗} = D_{2}ξ and g^{∗} = D_{2}η. Notice that ξ, η is a regular sequence in G. We have b ≥ a > d_{2} > 0, and μ_{G}(I^{∗}) = n ≥ 3. There exists a minimal homogeneous system {ξ_{1}, ξ_{2}, …, ξ_{n}} of generators of I^{∗} such that ξ_{1} = f^{∗} and ξ_{2} = g^{∗}, and c_{i} := deg ξ_{i} ≤ deg ξ_{i+1} := c_{i+1} for each i ≤ n − 1. However, the ideal I^{∗} may fail to be perfect, and it is possible to have D_{3} := GCD(ξ_{1}, ξ_{2}, ξ_{3}) = D_{2} as is illustrated in [GHK1, Example 1.6]. We prove in [GHK1, Theorem 1.2] that I^{∗} is perfect if n = 3. We also prove in [GHK1] that ξ_{3} = h^{∗}, where h has the form h = αf +βg ∈ I with o(α) = b−d_{2}, and o(β) = a−d_{2}, and that c_{3} := o(h) > a+b−d_{2}. Moreover, if q = σf +τg is such that q^{∗} ∉ (f^{∗}, g^{∗})G and (o)(σ) = b − d_{2}, then o(q) = o(h) and (f^{∗}, g^{∗}, h^{∗})G = (f^{∗}, g^{∗}, q^{∗})G. Thus the ideal (ξ_{1}, ξ_{2}, ξ_{3})G is independent of the choice of ξ_{3}. In the case where n ≥ 4, we also prove that c_{4} ≥ c_{3} + 2 [GHK1, Proposition 2.4]. However, examples shown to us by Craig Huneke and Lance Bryant show that it is possible to have c_{i+1} = c_{i} for i ≥ 4. This resolves a question mentioned in [GHK1, Discussion 2.5]. If I^{∗} is perfect, we prove in [GHK2] that c_{i+1} ≥ c_{i} + 2 for each i with 2 ≤ i ≤ n − 1. A question raised in [GHK2] that remains open is whether for I^{∗} perfect in this higher dimensional setting, does it follow that I^{∗} ⊆ (ξ_{1}/D_{2}, ξ_{2}/D_{2})G.

THE LEADING IDEAL OF A COMPLETE INTERSECTION 5

2. The Main Results

We record in Proposition 2.1 behavior of the Hilbert function with respect to an inverse of a transformation considered by Kothari and described in parts (1) and (2) of [GHK2, Corollary 2.5].

Proposition 2.1. Assume notation as in Setting 1.1 and let n = μ_{G}(I^{∗}). Let ξ_{1}, ξ_{2}, ⋯, ξ_{n} be a minimal homogeneous system of generators for I^{∗} satisfying conditions (1) and (2) in Theorem 1.2, and let c_{i} = deg ξ_{i} for 1 ≤ i ≤ n. Also assume that 𝔫 = (x, y) and x^{∗} ∤ f^{∗}. Let m be a positive integer and consider the ideal J_{m} := (f, x^{m}g)S. Then the following assertions hold true.

(1) μ_{G}(J_{m}^{∗}) = μ_{G}(I^{∗}).

(2) J_{m}^{∗} has ξ_{1}, X^{m}ξ_{2}, X^{m}ξ_{3}, …, X^{m}ξ_{n} as a minimal homogeneous system of generators, so the degree sequence for J_{m}^{∗} is (c′_{1}, c′_{2}, …, c′_{n}), where c′_{1} = c_{1} and c′_{i} = c_{i} + m for i ≥ 2.

(3) With D′_{i} = GCD(ξ_{1}, X^{m}ξ_{2}, …, X^{m}ξ_{i}) and d′_{i} = deg D′_{i}, we have D′_{i} = D_{i} and d′_{i} = d_{i} for 1 ≤ i ≤ n.

(4) H(gr_{𝔫}(S/J_{m}), λ) = \frac{∑_{i=2}^{n} λ^{d_{i}}(1 − λ^{d_{i−1}−d_{i}})(1 − λ^{m}λ^{c_{i}−d_{i}})}{(1 − λ)^{2}}.

(5) ℓ(S/J_{m}) − ℓ(S/I) = ma.

Since the residue field of S is infinite, with notation as in Setting 1.1 we may choose x, y so that 𝔫 = (x, y) and x^{∗} ∤ f^{∗} and y^{∗} ∤ f^{∗}. Thus it is possible to obtain the hypothesis of Theorem 2.2.

Theorem 2.2. Assume notation as in Setting 1.1 with n = μ_{G}(I^{∗}), a < b and 𝔫 = (x, y) such that x^{∗} ∤ f^{∗} and y^{∗} ∤ f^{∗}. Let ξ_{1}, ξ_{2}, ⋯, ξ_{n} be a minimal homogeneous system of generators for I^{∗} satisfying conditions (1) and (2) in Theorem 1.2 and let c_{i} = deg ξ_{i} for 1 ≤ i ≤ n. Let m be a positive integer and set

V_{m} = (y^{m}f, x^{m}(f + g))S.

Then the following assertions hold true.

(1) V_{m} is 𝔫-primary and μ_{G}(V_{m}^{∗}) = μ_{G}(I^{∗}) + 1.

(2) V_{m}^{∗} has Y^{m}ξ_{1}, X^{m}ξ_{1}, X^{m}Y^{m}ξ_{2}, X^{m}Y^{m}ξ_{3}, …, X^{m}Y^{m}ξ_{n} as a minimal homogeneous system of generators, so the degree sequence for V_{m}^{∗} is (c′_{1}, c′_{2}, …, c′_{n+1}), where c′_{1} = c_{1} + m, c′_{2} = c_{1} + m and c′_{i} = c_{i−1} + 2m for i ≥ 3.

6 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

(3) With D′_{i} = GCD(Y^{m}ξ_{1}, X^{m}ξ_{1}, X^{m}Y^{m}ξ_{2}, …, X^{m}Y^{m}ξ_{i−1}) and d′_{i} = deg D′_{i}, we have D′_{1} = Y^{m}D_{1}, and D′_{i} = D_{i−1} for 2 ≤ i ≤ n + 1. Thus d′_{1} = m + d_{1} and d′_{i} = d_{i−1} for 2 ≤ i ≤ n + 1.

(4) H(gr_{𝔫}(S/V_{m}), λ) = \frac{∑_{i=2}^{n+1} λ^{d′_{i}}(1 − λ^{d′_{i−1}−d′_{i}})(1 − λ^{c′_{i}−d′_{i}})}{(1 − λ)^{2}}

= λ^{a}(1 + λ + λ^{2} + ⋯ + λ^{m−1})^{2} + \frac{∑_{i=2}^{n} λ^{d_{i}}(1 − λ^{d_{i−1}−d_{i}})(1 − λ^{2m}λ^{c_{i}−d_{i}})}{(1 − λ)^{2}}.

(5) ℓ(S/V_{m}) − ℓ(S/I) = m(2a + m).

Proof. To show that V_{m} is 𝔫-primary, we observe that y^{m}f is not in any minimal prime P of x^{m}(f + g). Since a < b, X^{m}f^{∗} is the leading form of x^{m}(f + g). Since y^{∗} ∤ f^{∗}, we see that y ∉ P. Since (f, f + g)S = I and x^{∗} ∤ f^{∗}, we see that f ∉ P. Thus V_{m} is 𝔫-primary. Let V′_{m} = (f + g, y^{m}f)S. The part (2) of [GHK2, Corollary 2.5] implies that μ_{G}(I) = μ_{G}(V′_{m}) and the part (3) of [GHK2, Corollary 2.5] implies μ_{G}(V′_{m})+1 = μ_{G}(V_{m}). It also follows from Corollary 2.5 in [GHK2] that Y^{m}ξ_{1}, X^{m}ξ_{1}, X^{m}Y^{m}ξ_{2}, X^{m}Y^{m}ξ_{3}, …, X^{m}Y^{m}ξ_{n} is a minimal homogeneous system of generators for V_{m}^{∗}. The remaining assertions in Theorem 2.2 follow from this. □

In Theorem 2.3, we establish the existence of examples to show that every Hilbert series described in Theorem 1.3 is realizable as the Hilbert series of gr_{𝔪}(R) for some complete intersection ideal I = (f, g).

Theorem 2.3. Let a and b be positive integers with a ≤ b and consider a system consisting of

(1) An integer n with 2 ≤ n ≤ a + 1.

(2) A sequence (a = c_{1}, b = c_{2}, c_{3}, …, c_{n}) of integers.

(3) A sequence (a = d_{1}, d_{2}, …, d_{n} = 0) of integers such that

c_{i+1} − c_{i} > d_{i−1} − d_{i} > 0 for all i with 2 ≤ i ≤ n − 1. For each system satisfying these conditions, there exists an ideal I = (f, g) as in Setting 1.1 such that (ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, ξ_{3}, …, ξ_{n}) is a minimal set of homogeneous generators of I^{∗}, deg ξ_{i} = c_{i}, and d_{i} = deg D_{i}, where D_{i} = GCD(ξ_{1}, …, ξ_{i}), for each i with 1 ≤ i ≤ n.

Proof. The proof is by induction on a. If a = 1, then n = 2 and f = x^{a}, g = y^{b} shows the assertion holds in this case. Let a′ > 1 be an integer and assume that the assertion holds for all positive integers a < a′. Let (a′ = c′_{1}, b′ = c′_{2}, c′_{3}, …, c′_{n+1}) and (a′ = d′_{1}, d′_{2}, …, d′_{n+1} = 0) be sequences of n + 1 integers such that

c′_{i+1} − c′_{i} > d′_{i−1} − d′_{i} > 0 for all i with 2 ≤ i ≤ n.

Notice that d′_{i−1} − d′_{i} > 0 for all i with 2 ≤ i ≤ n and a′ = ∑_{i=2}^{n+1}(d′_{i−1} − d′_{i}) implies n + 1 ≤ a′ + 1. Let e := c′_{2} − c′_{1}. By Proposition 2.1, the system consisting of the integer n + 1 and the sequences (c′_{1}, c′_{2}, …, c′_{n+1}) and (a′ = d′_{1}, d′_{2}, …, d′_{n+1} = 0) is realizable if and only if the system consisting of n + 1 and the sequences (c′_{1}, c′_{2} − e, …, c′_{n+1} − e) and (a′ = d′_{1}, d′_{2}, …, d′_{n+1} = 0) is realizable. Thus we may assume that c′_{1} = c′_{2} = a′.

Let m = d′_{1} − d′_{2} and a = a′ − m. Then a = d′_{2}. Consider the system consisting of the positive integer n and the sequences (a = c_{1}, c_{2}, …, c_{n}) and (a = d_{1}, …, d_{n}), where c_{i} = c′_{i+1} − 2m and d_{i} = d′_{i+1} for each i with 2 ≤ i ≤ n. We have

c_{i+1} − c_{i} > d_{i−1} − d_{i} > 0 for all i with 2 ≤ i ≤ n − 1.

Also a = ∑_{i=2}^{n}(d_{i−1} − d_{i}) implies n ≤ a + 1. By our inductive hypothesis the system consisting of the integer n and the sequences (a = c_{1}, c_{2}, …, c_{n}) and (a = d_{1}, …, d_{n} = 0) is realizable. Moreover, c_{2} = c′_{3} and c′_{3} − a′ > m and a′ = a + m implies c_{2} > a. Therefore by Theorem 2.2 the system consisting of n + 1 and the sequences (c′_{1}, c′_{2} −e, …, c′_{n+1} −e) and (a′ = d′_{1}, d′_{2}, …, d′_{n+1} = 0) is realizable. This completes the proof of Theorem 2.3. □

3. Remarks and Examples

Remark 3.1. With notation and hypothesis as in Theorems 1.2 and 1.3, it follows from Theorem 1.3 that the Hilbert series H(gr_{𝔪}(R), λ) = ∑_{i=0}^{∞} dim_{k}[gr_{𝔪}(R)]_{i}λ^{i} is uniquely determined by the degree sequence (c_{1}, c_{2}, …, c_{n}) together with the sequence (a = d_{1} > d_{2} > d_{3} > ⋯ > d_{n} = 0). Notice also that dim_{k}[gr_{𝔪}(R)]_{c_{i}} = d_{i−1} − 1, for each i with 2 ≤ i ≤ n. Usually the sequence (c_{1}, c_{2}, ⋯, c_{n}) does not uniquely determine the Hilbert series of gr_{𝔪}(R), but Theorem 1.3 implies the following.

(1) n = μ_{G}(I^{∗}) ≤ d_{n−1} + (n − 1) ≤ ⋯ ≤ d_{3} + 3 ≤ d_{2} + 2 ≤ a + 1.

(2) n = μ_{G}(I^{∗}) = a + 1 ⟺ d_{i−1} − d_{i} = 1 for each i with 2 ≤ i ≤ a + 1. Therefore, in this case, d_{i} = n − i for each i with 1 ≤ i ≤ n, and the Hilbert series of gr_{𝔪}(R) is uniquely determined by the sequence (c_{1}, c_{2}, ⋯, c_{n}).

8 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

(3) If μ_{G}(I^{∗}) = a + 1, then e(gr_{𝔪}(R)) ≥ \frac{3a^{2}−a}{2}, and equality holds ⟺ a = b and c_{i+1} − c_{i} = 2 for each i with 2 ≤ i ≤ a. In this case, the degree sequence of I^{∗} is (a, a, a + 2, a + 4, a + 6, ⋯, a + (a − 1)2).

Example 3.2. We describe examples of parameter ideals I = (f, g)S for which μ_{G}(I^{∗}) = 3. Let (a, b, c) be a sequence of integers such that 2 ≤ a ≤ b < b + 2 ≤ c. If (a, b, c) is the degree sequence of I^{∗}, where I = (f, g), then Theorem 1.3 implies that d := d_{2} = deg GCD(f^{∗}, g^{∗}) must satisfy c − b > a − d > 0.

Let

f = x^{d}y^{a−d} and g = x^{b} + y^{c+d−a}.

Notice that c + d − a > b. We have

I^{∗} = (x^{∗}^{d}y^{∗}^{a−d}, x^{∗}^{b}, y^{∗}^{c}) = I_{2} (\begin{matrix}x^{∗}^{b−d} & y^{∗}^{a−d} & 0 \\ y^{∗}^{c+d−a} & 0 & x^{∗}^{d} \end{matrix})

Remark 3.3. Let I = (f, g)S be a parameter ideal such that μ_{G}(I^{∗}) = 3. By Theorem 1.3, the Hilbert series of gr_{𝔪}(R) is uniquely determined by the degree sequence (a, b, c) together with the integer d := d_{2} = deg GCD(f^{∗}, g^{∗}); moreover, we must have c − b > a − d > 0. Example 3.2 demonstrates that each system (a, b, c) and d with c − b > a − d > 0 is realizable. For integers (a, b, c) with 2 ≤ a ≤ b < b + 2 ≤ c, the possible values for d are constrained by the conditions d ≤ a − 1 and d > min{0, a + b − c}. Thus the number of possible values for d, and hence the number of different Hilbert series associated with the degree sequence (a, b, c), is min{a − 1, c − b − 1}.

Example 3.4 illustrates Theorem 2.3 in the case where μ_{G}(I^{∗}) = 4.

Example 3.4. With the notation of Theorem 1.3, if I^{∗} has degree sequence (4, 5, 8, 11), then the possibilities for the sequence (d_{1}, d_{2}, d_{3}, d_{4}) are

(4, 3, 2, 0), (4, 3, 1, 0), (4, 2, 1, 0).

(1) This gives, for appropriate ideals I^{∗}, the following three Hilbert series for G/I^{∗}.

(i) Q_{G/I^{∗}}(t) = 1 + 2t + 3t^{2} + 4t^{3} + 4t^{4} + 3t^{5} + 3t^{6} + 3t^{7} + 2t^{8} + 2t^{9} + 2t^{10} + t^{11},
(ii) Q_{G/I^{∗}}(t) = 1 + 2t + 3t^{2} + 4t^{3} + 4t^{4} + 3t^{5} + 3t^{6} + 3t^{7} + 2t^{8} + t^{9} + t^{10},
(iii) Q_{G/I^{∗}}(t) = 1 + 2t + 3t^{2} + 4t^{3} + 4t^{4} + 3t^{5} + 2t^{6} + 2t^{7} + t^{8} + t^{9} + t^{10}.

corresponding to the sequences (4, 3, 2, 0), (4, 3, 1, 0), (4, 2, 1, 0).

THE LEADING IDEAL OF A COMPLETE INTERSECTION 9

(2) Using Proposition 2.1 and Theorem 2.2, we may obtain from the degree sequence (4, 5, 8, 11) of I^{∗}, either the degree sequence (3, 3, 6) or (2, 2, 5). Indeed, (3, 3, 6) is obtained by

(4, 5, 8, 11) ⟶^{m=1 in Prop 2.1} (4, 4, 7, 10) ⟶^{m=1 in Thm 2.2} (3, 5, 8) ⟶^{m=2 in Prop 2.1} (3, 3, 6),

and (2, 2, 5) is obtained by

(4, 5, 8, 11) ⟶^{m=1 in Prop 2.1} (4, 4, 7, 10) ⟶^{m=2 in Thm 2.2} (2, 3, 6) ⟶^{m=1 in Prop 2.1} (2, 2, 5).

The sequence (3, 3, 6) is the degree sequences associated with (x^{3} + y^{5}, x^{2}y)S and (2, 2, 5) is the degree sequence associated with (x^{2} + y^{4}, xy)S.

(i) Let I1^{′} = (f_{1}, g_{1}), where f_{1} = x^{3} + y^{5}, g_{1} = x^{2}y. Since y^{∗} ∤ f_{1}^{∗}, Proposition 2.1 implies that J1^{′} = (f_{1}, y^{2}g_{1}) has degree sequence (3, 5, 8). Let n = (x + y, y). Then (x^{∗}+y^{∗}) ∤ f_{1}^{∗} and y^{∗} ∤ f_{1}^{∗}, so Theorem 2.2 implies that V1^{′} = (yf_{1}, (x+y)(f_{1}+y^{2}g_{1})) = (F_{1}, G_{1}) has degree sequence (4, 4, 7, 10). Since y^{∗} ∤ G_{1}^{∗}, Proposition 2.1 implies that I_{1} = (yF_{1}, G_{1}) = (y^{2}f_{1}, (x + y)(f_{1} + y^{2}g_{1}) has degree sequence (4, 5, 8, 11). Also the descending sequence (d_{1}, d_{2}, d_{3}, d_{4}) of I_{1}^{∗} is (4, 3, 2, 0). Hence the Hilbert series of G/I_{1}^{∗} is (i) of part (1). (ii) Let I2^{′} = (f_{2}, g_{2}), where f_{2} = x^{3} + y^{4}, g_{2} = xy^{2}. Since y^{∗} ∤ f_{2}^{∗}, Proposition 2.1 implies that J2^{′} = (f_{2}, y^{2}g_{2}) has degree sequence (3, 5, 8). Let n = (x + y, y). Then (x^{∗}+y^{∗}) ∤ f_{2}^{∗} and y^{∗} ∤ f_{2}^{∗}, so Theorem 2.2 implies that V2^{′} = (yf_{2}, (x+y)(f_{2}+y^{2}g_{2})) = (F_{2}, G_{2}) has degree sequence (4, 4, 7, 10). Since y^{∗} ∤ G_{2}^{∗}, Proposition 2.1 implies that I_{2} = (yF_{2}, G_{2}) = (y^{2}f_{2}, (x + y)(f_{2} + y^{2}g_{2}) has degree sequence (4, 5, 8, 11). The descending sequence (d_{1}, d_{2}, d_{3}, d_{4}) of I_{2}^{∗} is (4, 3, 1, 0). Hence the Hilbert series of G/I_{2}^{∗} is (ii) of part (1). (iii) Let I3^{′} = (f_{3}, g_{3}), where f_{3} = x^{2} + y^{4}, g_{3} = xy. Since y^{∗} ∤ f_{3}^{∗}, Proposition 2.1 implies that J3^{′} = (f_{3}, yg_{3}) has degree sequence (2, 3, 6). Let n = (x + y, y). Then (x^{∗} + y^{∗}) ∤ f_{3}^{∗} and y^{∗} ∤ f_{3}^{∗}, so Theorem 2.2 implies that V3^{′} = (y^{2}f_{3}, (x + y)^{2}(f_{3} + yg_{3})) = (F_{3}, G_{3}) has degree sequence (4, 4, 7, 10). Since y^{∗} ∤ G_{3}^{∗}, Proposition 2.1 implies that I_{3} = (yF_{3}, G_{3}) = (y^{3}f_{3}, (x + y)^{2}(f_{3} + yg_{3}) has degree sequence (4, 5, 8, 11). The descending sequence (d_{1}, d_{2}, d_{3}, d_{4}) of I_{3}^{∗} is (4, 2, 1, 0). Hence the Hilbert series of G/I_{3}^{∗} is (iii) of part (1).

Example 3.5 illustrates Theorem 2.3 in the case where μ_{G}(I^{∗}) = 5.

10 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

Example 3.5. Assume notation as in Theorems 1.2 and 1.3 and also assume that char(k) ≠ 2. If I^{∗} has degree sequence (6, 7, 12, 15, 18), then the possibilities for the sequence (d_{1}, d_{2}, d_{3}, d_{4}, d_{5}) are

\begin{matrix}(6, 5, 4, 3, 0), (6, 5, 4, 2, 0), (6, 5, 3, 2, 0), (6, 5, 3, 1, 0), \\ (6, 4, 3, 2, 0), (6, 4, 3, 1, 0), (6, 4, 2, 1, 0), (6, 3, 2, 1, 0). \end{matrix}

The following eight parameter ideals I_{i} = (f_{i}, g_{i})S have associated degree sequence (6, 7, 12, 15, 18) and each of the eight possible sequences (d_{1}, d_{2}, d_{3}, d_{4}, d_{5}).

(i) I_{1} = (yy(x + y)(f_{1} + y^{2}g_{1}), (x + 2y)(x + y)(f_{1} + y^{2}g_{1} + y^{4}yf_{1})) where f_{1} = x^{4} + y^{6} and g_{1} = x^{3}y,

(ii) I_{2} = (yy(x + y)(f_{2} + y^{2}g_{2}), (x + 2y)(x + y)(f_{2} + y^{2}g_{2} + y^{4}yf_{2})), where f_{2} = x^{4} + y^{5} and g_{2} = x^{2}y^{2},

(iii) I_{3} = (yy(x + y)^{2}(f_{3} + yg_{3}), (x + 2y)(x + y)^{2}(f_{3} + yg_{3} + y^{4}y^{2}f_{3})), where f_{3} = x^{3} + y^{5} and g_{3} = x^{2}y,

(iv) I_{4} = (yy^{2}(x + y)(f_{4} + y^{2}g_{4}), (x + 2y)^{2}(x + y)(f_{4} + y^{2}g_{4} + y^{3}yf_{4})), where f_{4} = x^{3} + y^{5} and g_{4} = x^{2}y,

(v) I_{5} = (yy(x + y)^{2}(f_{5} + yg_{5}), (x + 2y)(x + y)^{2}(f_{5} + y^{2}g_{5} + y^{4}y^{2}f_{5})), where f_{5} = x^{3} + y^{4} and g_{5} = xy^{2},

(vi) I_{6} = (yy^{2}(x + y)(f_{6} + y^{2}g_{6}), (x + 2y)^{2}(x + y)(f_{6} + y^{2}g_{6} + y^{3}yf_{6})), where f_{6} = x^{3} + y^{4} and g_{6} = xy^{2},

(vii) I_{7} = (yy^{2}(x + y)^{2}(f_{7} + yg_{7}), (x + 2y)^{2}(x + y)^{2}(f_{7} + yg_{7} + y^{3}yf_{7})), where f_{7} = x^{2} + y^{4} and g_{7} = xy,

(viii) I_{8} = (yy^{3}(x + y)(f_{8} + y^{2}g_{8}), (x + 2y)^{3}(x + y)(f_{8} + y^{2}g_{8} + y^{2}yf_{8})), where f_{8} = x^{2} + y^{4} and g_{8} = xy.

To obtain these parameter ideals we reason as follows. Using Proposition 2.1 and Theorem 2.2, we may obtain from the degree sequence (6, 7, 12, 15, 18) of I^{∗} one of the following three degree sequences (4, 4, 7), (3, 3, 6), or (2, 2, 5). Indeed, (4, 4, 7) is

THE LEADING IDEAL OF A COMPLETE INTERSECTION 11

obtained by

(6, 7, 12, 15, 18) ⟶^{m=1 in Prop 2.1} (6, 6, 11, 14, 17) ⟶^{m=1 in Thm 2.2} (5, 9, 12, 15)
⟶^{m=4 in Prop 2.1} (5, 5, 8, 11) ⟶^{m=1 in Thm 2.2} (4, 6, 9) ⟶^{m=2 in Prop 2.1} (4, 4, 7).

The sequence (3, 3, 6) is obtained in two ways:

(6, 7, 12, 15, 18) ⟶^{m=1 in Prop 2.1} (6, 6, 11, 14, 17) ⟶^{m=1 in Thm 2.2} (5, 9, 12, 15)
⟶^{m=4 in Prop 2.1} (5, 5, 8, 11) ⟶^{m=2 in Thm 2.2} (3, 4, 7) ⟶^{m=1 in Prop 2.1} (3, 3, 6).

and

(6, 7, 12, 15, 18) ⟶^{m=1 in Prop 2.1} (6, 6, 11, 14, 17) ⟶^{m=2 in Thm 2.2} (4, 7, 10, 13)
⟶^{m=3 in Prop 2.1} (4, 4, 7, 10) ⟶^{m=1 in Thm 2.2} (3, 5, 8) ⟶^{m=2 in Prop 2.1} (3, 3, 6).

The sequence (2, 2, 5) is also obtained in two ways:

(6, 7, 12, 15, 18) ⟶^{m=1 in Prop 2.1} (6, 6, 11, 14, 17) ⟶^{m=2 in Thm 2.2} (4, 7, 10, 13)
⟶^{m=3 in Prop 2.1} (4, 4, 7, 10) ⟶^{m=2 in Thm 2.2} (2, 3, 6) ⟶^{m=1 in Prop2.1} (2, 2, 5).

and

(6, 7, 12, 15, 18) ⟶^{m=1 in Prop 2.1} (6, 6, 11, 14, 17) ⟶^{m=3 in Thm 2.2} (3, 5, 8, 11)
⟶^{m=2 in Prop 2.1} (3, 3, 6, 9) ⟶^{m=1 in Thm 2.2} (2, 4, 7) ⟶^{m=2 in Prop 2.1} (2, 2, 5).

We describe a procedure for obtaining the parameter ideal I_{8}. Let I8^{′′} = (f_{8}, g_{8}), where f_{8} = x^{2} + y^{4}, g_{8} = xy. Then I8^{′′} has degree sequence (2, 2, 5). Since y^{∗} ∤ f_{8}^{∗}, Proposition 2.1 implies that J8^{′′} = (f_{8}, y^{2}g_{8}) has degree sequence (2, 4, 7). Let n = (x + y, y). Then (x^{∗} + y^{∗}) ∤ f_{8}^{∗} and y^{∗} ∤ f_{8}^{∗}, so Theorem 2.2 implies that V8^{′′} = (yf_{8}, (x + y)(f_{8} + y^{2}g_{8})) = (F8^{′}, G8^{′}) has degree sequence (3, 3, 6, 9). Since y^{∗} ∤ G8^{′^{∗}}, Proposition 2.1 implies that J8^{′} = (y^{2}F8^{′}, G8^{′}) has degree sequence (3, 5, 8, 11). Since n = (x + 2y, y) = (x + y, y) = (x, y) and (x + 2y)^{∗} ∤ G8^{′^{∗}} and y^{∗} ∤ G8^{′^{∗}}, Theorem 2.2 implies that V8^{′} = (y^{3}G8^{′}, (x + 2y)^{3}(G8^{′} + y^{2}F8^{′})) = (F_{8}, G_{8}) has degree sequence (6, 6, 11, 14, 17). Since y^{∗} ∤ G_{8}^{∗}, Proposition 2.1 implies that I_{8} = (yF_{8}, G_{8}) =

(y^{4}G8^{′}, (x+2y)^{3}(G8^{′}+y^{2}F8^{′}) = (y^{4}(x+y)(f_{8}+y^{2}g_{8}), (x+2y)^{3}(x+y)(f_{8}+y^{2}g_{8}+y^{3}f_{8})),

has degree sequence (6, 7, 12, 15, 18).

Similar reasoning is used to obtain the parameter ideals I_{i}, for 1 ≤ i ≤ 7.

References

[Br] J. Briançon, Description de Hilb^{n}C{x, y}, Invent. Math. 41 (1977), 45-89.

[GHK1] S. Goto, W. Heinzer, and M.-K. Kim, The leading ideal of a complete intersection of height two, J. Algebra, 298 (206), 238-247.

[GHK2] S. Goto, W. Heinzer, and M.-K. Kim, The leading ideal of a complete intersection of height two, II, J. Algebra (to appear).

[Ia1] A. Iarrobino, Tangent cone of a Gorenstein singularity, in Proceedings of the conference in Algebraic Geometry, Berlin 1985, H. Kurke and M. Roczen, eds., Teubnertexte zur Math. 92, Teubner, Leipzig, 1986, 163-176.

[Ia2] A. Iarrobino, Associated Graded Algebra of a Gorenstein Artin Algebra, Memoirs Amer. Math. Soc., 107, num. 514, 1994.

[K] S. C. Kothari, The local Hilbert function of a pair of plane curves, Proc. Amer. Math. Soc., 72 (1978), 439-442.

[M] F. S. Macaulay, On a method of dealing with the intersection of plane curves, Trans. Amer. Math. Soc., 5 (1904), 385-410.

Department of Mathematics, School of Science and Technology Meiji University, 214-8571 Japan

   E-mail address: goto@math.meiji.ac.jp

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

   E-mail address: heinzer@math.purdue.edu

Department of Mathematics, Sungkyunkwan University Jangangu Suwon 440-746, Korea

   E-mail address: mkkim@skku.edu