An application of Liaison theory to zero-dimensional schemes

TL;DR

This paper applies Liaison theory to characterize the Cayley-Bacharach property of zero-dimensional schemes in projective space, extending previous results for rational points and analyzing related algebraic invariants.

Contribution

It extends the characterization of the Cayley-Bacharach property to arbitrary zero-dimensional schemes using Liaison theory and investigates bounds on the Hilbert function and regularity index of the Dedekind different.

Findings

Extended Cayley-Bacharach characterization to all zero-dimensional schemes

Bounded the Hilbert function of the Dedekind different

Analyzed the regularity index of the Dedekind different

Abstract

Given a 0-dimensional scheme X in a n-dimensional projective space P^n_K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of K-rational points given in [7]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of X when X has the Cayley-Bacharach property.

Equations

$$\mathrm{ri}(M)=\min \{i\in \mathbb{Z}\mid {\mathrm{HF}}_M(j)={\mathrm{HP}}_M(j)\text{for all}j\ge i\}$$

$$1={\mathrm{HF}}_{\mathbb{X}}(0)<{\mathrm{HF}}_{\mathbb{X}}(1)<\cdots <{\mathrm{HF}}_{\mathbb{X}}(r_{\mathbb{X}}-1)<\mathrm{deg}(\mathbb{X})$$

$$\tilde{ı}:R_{\mathbb{X}}\to Q^h(R_{\mathbb{X}})\cong \prod _{j=1}^s{\mathcal{O}}_{\mathbb{X},p_j}[T_j,T_j^{-1}]$$

$$\mu (a):=\min \{i\in \mathbb{N}\mid (0,\dots ,0,a{T}_j^i,0,\dots ,0)\in \tilde{ı}(R_{\mathbb{X}})\}.$$

$$f_{j{k}_j}^{\ast }:={\tilde{ı}}^{-1}((0,\dots ,0,e_{j{k}_j}{s}_j{T}_j^{\mu (e_{j{k}_j}{s}_j)},0,\dots ,0))$$

$${\mu }_{{\mathbb{X}}^{\prime }/\mathbb{X}}:=\max \{\mathrm{deg}(f_{j{k}_j}^{\ast })\mid k_j=1,\dots ,{\varkappa }_j\}$$

$$(I_{\mathbb{X}^{\prime}/\mathbb{X}})_{i}=\big{\langle}\,x_{0}^{i-\deg(f^{*}_{jk_{j}})}f^{*}_{jk_{j}}\,\mid\,\deg(f^{*}_{jk_{j}})\leq i\,\big{\rangle}_{K}$$

$$\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{W}}}(i)=\deg(\mathbb{X})-\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{W}}-i-1)\quad\mbox{for all $i\in\mathbb{Z}$.}$$

$$I_{\mathbb{Y}/\mathbb{W}}({\alpha }_{\mathbb{Y}/\mathbb{W}})$$

$$\cong {\mathrm{Hom}}_{R_{\mathbb{W}}}(R_{\mathbb{W}}/I_{\mathbb{X}/\mathbb{W}},R_{\mathbb{W}}(r_{\mathbb{W}}))(-r_{\mathbb{X}})$$

$$\cong {\mathrm{Hom}}_{R_{\mathbb{W}}}(R_{\mathbb{X}},{\mathrm{Hom}}_{K[x_0]}(R_{\mathbb{W}},K[x_0]))(-r_{\mathbb{X}})$$

$$\cong {\mathrm{Hom}}_{K[x_0]}(R_{\mathbb{X}},K[x_0])(-r_{\mathbb{X}}).$$

$$({\mathrm{Ann}}_{{\overline{R}}_{\mathbb{W}}}(({\overline{I}}_{\mathbb{Y}/\mathbb{W}}{)}_{{\alpha }_{\mathbb{Y}/\mathbb{W}}+(r_{\mathbb{X}}-d)}){)}_d$$

$$=({\mathrm{Ann}}_{{\overline{R}}_{\mathbb{W}}}({\overline{I}}_{\mathbb{Y}/\mathbb{W}}){)}_d=({\overline{I}}_{\mathbb{X}/\mathbb{W}}{)}_d.$$

$${\mathcal{O}}_{\mathbb{X},p_j}=\{\begin{array}{ll}{\mathcal{O}}_{\mathbb{W},p_j}& \text{mbox}forj=1,\dots ,s,\\ {\mathcal{O}}_{\mathbb{W},p_j}/{\mathfrak{q}}_j& \text{mbox}forj=s+1,\dots ,t,\\ ⟨0⟩& \text{mbox}forj=t+1,\dots ,u.\end{array}$$

$$\theta (I_{\mathbb{X}/\mathbb{W}})=\theta ((I_{\mathbb{X}/\mathbb{W}}{)}_{r_{\mathbb{W}}})=⟨0⟩\times \cdots \times ⟨0⟩\times {\mathfrak{q}}_{s+1}\times \cdots \times {\mathfrak{q}}_t\times {\mathcal{O}}_{\mathbb{W},p_{t+1}}\times \cdots \times {\mathcal{O}}_{\mathbb{W},p_u}$$

$$\Lambda :={\mathcal{O}}_{\mathbb{W},p_1}\times \cdots \times {\mathcal{O}}_{\mathbb{W},p_s}\times {\mathrm{Ann}}_{{\mathcal{O}}_{\mathbb{W},p_{s+1}}}({\mathfrak{q}}_{s+1})\times \cdots \times {\mathrm{Ann}}_{{\mathcal{O}}_{\mathbb{W},p_t}}({\mathfrak{q}}_t)\times ⟨0⟩\times \cdots \times ⟨0⟩.$$

$$\theta (I_{{\mathbb{X}}^{\prime }/\mathbb{W}})$$

$$\theta (I_{{\mathbb{Y}}^{\prime }/\mathbb{W}})$$

$$\times {\mathrm{Ann}}_{{\mathcal{O}}_{\mathbb{W},p_t}}({\mathfrak{q}}_t)\times \{0\}\times \cdots \times \{0\}.$$

$$\theta (I_{{\mathbb{X}}^{\prime }/\mathbb{W}})$$

$$\theta (I_{{\mathbb{Y}}^{\prime }/\mathbb{W}})$$

$$\times {\mathrm{Ann}}_{{\mathcal{O}}_{\mathbb{W},p_t}}({\mathfrak{q}}_t)\times \{0\}\times \cdots \times \{0\}.$$

$$\deg_{\mathbb{X}}(p_{j}):=\min\big{\{}\,\mu_{\mathbb{X}^{\prime}/\mathbb{X}}\;\big{|}\;\mathbb{X}^{\prime}\ \textrm{is a maximal $p_{j}$-subscheme of $\mathbb{X}$}\,\big{\}},$$

$${\dim }_K(I_{{\mathbb{X}}^{\prime }/\mathbb{X}}{)}_d<{\varkappa }_j.$$

$${\dim }_K{J}_{r_{\mathbb{W}}-1+i}\ge {\dim }_K⟨(G_{j1}{H}_j{)}_{p_j},\dots ,(G_{j{\varkappa }_j}{H}_j{)}_{p_j}{⟩}_K={\varkappa }_j$$

$$X_i{G}_{jl}+I_{\mathbb{X}}=\sum _{k_j=1}^{{\varkappa }_j}{c}_{j{k}_{j}l}{X}_0{G}_{j{k}_j}+I_{\mathbb{X}}$$

$$X_i{G}_{jl}{H}_j+I_{\mathbb{W}}=\sum _{k_j=1}^{{\varkappa }_j}{c}_{j{k}_{j}l}{X}_0{G}_{j{k}_j}{H}_j+I_{\mathbb{W}},$$

$$I_{{\mathbb{Y}}^{\prime }}=⟨X_0^2-\frac{1}{4}{X}_1^2-\frac{1}{2}{X}_0{X}_2-\frac{1}{4}{X}_1{X}_2,X_0{X}_1{X}_2+\frac{1}{2}{X}_1^2{X}_2,X_0{X}_2^2+\frac{1}{4}{X}_1{X}_2^2-\frac{1}{2}{X}_2^3⟩.$$

$${\mathrm{HF}}_{{\omega }_{R_{\mathbb{X}}}}(i)=\mathrm{deg}(\mathbb{X})-{\mathrm{HF}}_{\mathbb{X}}(-i)\text{mbox}foralli\in \mathbb{Z}.$$

$$\overline{\phi }(x_0(R_{\mathbb{X}}{)}_d)=\phi (x_0(R_{\mathbb{X}}{)}_d)=x_0\phi ((R_{\mathbb{X}}{)}_d)\subseteq x_0(K[x_0]{)}_{-1}=⟨0⟩.$$

$$\Big{\{}\,x_{0}^{i-d-1}g_{1},...,x_{0}^{i-d-1}g_{\scriptscriptstyle\sum\limits_{0\leq k\leq d+1}h_{k}},...,g_{\scriptscriptstyle\sum\limits_{0\leq k<i}h_{k}+1},...,g_{\scriptscriptstyle\sum\limits_{0\leq k\leq i}h_{k}}\,\Big{\}}$$

$$(R_{\mathbb{X}}{)}_i=⟨x_0^{i-d-1}{g}_1,...,x_0^{i-d-1}{g}_{\sum _{0\le k\le d+1}{h}_k},...,x_0^{i-r_{\mathbb{X}}}{g}_{\sum _{0\le k<r_{\mathbb{X}}}{h}_k+1},...,x_0^{i-r_{\mathbb{X}}}{g}_{\sum _{0\le k\le r_{\mathbb{X}}}{h}_k}{⟩}_K$$

$$f=\sum _{1\le j\le \sum _{0\le k\le d+1}{h}_k}{a}_j{x}_0^{i-d-1}{g}_j+\cdots +\sum _{\sum _{0\le k<r_{\mathbb{X}}}{h}_k+1\le j\le \sum _{0\le k\le r_{\mathbb{X}}}{h}_k}{a}_j{x}_0^{i-r_{\mathbb{X}}}{g}_j$$

Full text

1. Introduction

The theory of Liaison has been used very extensively in the literature as a tool to study projective varieties in the $n$-dimensional projective space $\mathbb{P}^{n}_{K}$. The initial idea was to start with a projective variety, and look at its residual variety in a complete intersection. Since complete intersections are well understood in some sense, one can get information about the variety from its residual variety or vice versa, and so it would be easier to pass to a “simpler” variety instead of considering a complicated one. This idea has been also generalized by allowing links by arithmetically Gorenstein schemes (see, e.g.,[22]). Currently, Liaison theory is an area of active research [2, 3, 4, 5, 7, 12, 23, 24, 25], and has many useful applications, for instance, constructing interesting projective varieties [2, 24, 25], or computing invariants and establishing properties of projective varieties [3, 4, 5, 9].

In this paper we are interested in applying the theory of Liaison to investigate the geometrical structure of 0-dimensional subschemes of the $n$-dimensional projective space $\mathbb{P}^{n}_{K}$ over an arbitrary field $K$. This approach was introduced by Geramita et al. [7] in their study of finite sets of $K$-rational points with the Cayley-Bacharach property. Classically, a finite set of $K$-rational points $\mathbb{X}$ in $\mathbb{P}^{n}_{K}$ is called a Cayley-Bachrach scheme if any hypersurface of degree less than the regularity index of the coordinate ring of $\mathbb{X}$ which contains all points of $\mathbb{X}$ but one automatically contains the last point. One of main results of [7] is stated as follows:

Theorem 1.1**.**

Let $\mathbb{W}$ be a set of points in $\mathbb{P}^{n}_{K}$ which is a complete intersection, let $\mathbb{X}\subseteq\mathbb{W}$, let $\mathbb{Y}=\mathbb{W}\setminus\mathbb{X}$, and let $I_{\mathbb{W}}$, $I_{\mathbb{X}}$ and $I_{\mathbb{Y}}$ denote the homogeneous vanishing ideals of $\mathbb{W},\mathbb{X}$ and $\mathbb{Y}$ in $P=K[X_{0},...,X_{n}]$, respectively. Set $\alpha_{\mathbb{Y}/\mathbb{W}}=\min\{i\in\mathbb{N}\mid(I_{\mathbb{Y}}/I_{\mathbb{W}})_{i}\neq\langle 0\rangle\}$. Then the following conditions are equivalent.

1. (a)

$\mathbb{X}$* is a Cayley-Bachrach scheme.* 2. (b)

A generic element of $(I_{\mathbb{Y}})_{\alpha_{\mathbb{Y}/\mathbb{W}}}$ does not vanish at any point of $\mathbb{X}$. 3. (c)

We have $I_{\mathbb{W}}:(I_{\mathbb{Y}})_{\alpha_{\mathbb{Y}/\mathbb{W}}}=I_{\mathbb{X}}$.

This result nicely leads to an efficient algorithm for checking whether a given set $\mathbb{X}$ is a Cayley-Bacharach scheme. Later investigations of the Cayley-Bacharach property have included the work of Fouli, Polini, and Ulrich [6], Robbiano [17], Gold, Little, and Schenck [8], and Guardo [10]. Moreover, this property has also been extended for 0-dimensional schemes in $\mathbb{P}^{n}_{K}$ (see [14, 15, 16, 21]). When $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ is a 0-dimensional scheme over an algebraically closed field $K$, Robbiano and the first author [17] considered subschemes of $\mathbb{X}$ of degree $\deg(\mathbb{X})-1$ to show that the conditions (a) and (c) of Theorem 1.1 are still equivalent. However, we get no further information for a generalization of condition (b) in this case. It is worth noting here that if $K$ is not algebraically closed then the scheme $\mathbb{X}$ may have no subschemes of degree $\deg(\mathbb{X})-1$. For example, the 0-dimensional scheme $\mathbb{X}={\mathcal{Z}}(2X_{0}^{4}+X_{0}^{2}X_{1}^{2}-X_{1}^{4})\subseteq\mathbb{P}^{1}_{\mathbb{Q}}$ is of degree 4, but it has no subscheme of degree 3.

Our focus in this paper is to look at an extension of the Cayley-Bacharach property and to generalize the above theorem for 0-dimensional schemes $\mathbb{X}$ in $\mathbb{P}^{n}_{K}$ over an arbitrary field $K$. In particular, we will look closely at the natural question whether conditions (a) and (b) of the above theorem are equivalent for our more general setting. Our approach is to use the notion of maximal $p_{j}$-subschemes of $\mathbb{X}$ which are introduced and studied in the papers [13, 14]. Also, we discuss a characterization of the Cayley-Bacharach property of degree $d$ with $d\in\mathbb{N}$ in terms of the canonical module of the coordinate ring of $\mathbb{X}$ and apply this result to bound the Hilbert function of the Dedekind different of $\mathbb{X}$ and determine its regularity index in some special cases.

This paper is structured as follows. In Section 2, we introduce the relevant information about Hilbert function, maximal $p_{j}$-subschemes, standard set of separators, and Liaison techniques. Especially, we give an explicit description of the residual scheme in a 0-dimensional arithmetically Gorenstein scheme of a maximal $p_{j}$-subscheme of $\mathbb{X}$. In Section 3, we prove the generalization of the results mentioned above (see Theorems 3.5 and 3.8). We also give Example 3.7 to show that the condition (b) in Theorem 1.1 is, in general, only a sufficient condition, not a necessary condition, for $\mathbb{X}$ being a Cayley-Bacharach scheme. In the final section, we characterize the Cayley-Bacharach property of degree $d$ using the canonical module of the coordinate ring of $\mathbb{X}$, and then look at the Hilbert function of the Dedekind different of $\mathbb{X}$ and its regularity index when $\mathbb{X}$ has the Cayley-Bacharach property of degree $d$.

All examples in this paper were calculated by using the computer algebraic system ApCoCoA (see [1]).

2. Basic Facts and Notation

Throughout the paper, we work over an arbitrary field $K$. The $n$-dimensional projective space over $K$ is denoted by $\mathbb{P}^{n}_{K}$ and its homogeneous coordinate ring is the polynomial ring $P=K[X_{0},...,X_{n}]$ equipped with the standard grading. Our object of interest is a 0-dimensional subscheme $\mathbb{X}$ of $\mathbb{P}^{n}_{K}$. Its homogeneous vanishing ideal in $P$ is denoted by $I_{\mathbb{X}}$ and its homogeneous coordinate ring is given by $R_{\mathbb{X}}=P/I_{\mathbb{X}}.$ The set of closed points of $\mathbb{X}$ is called the support of $\mathbb{X}$ and is denoted by $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$. We always assume that $\operatorname{Supp}(\mathbb{X})\cap{\mathcal{Z}}(X_{0})=\emptyset$. Under this assumption, the image $x_{0}$ of $X_{0}$ in $R_{\mathbb{X}}$ is a non-zerodivisor, and hence $R_{\mathbb{X}}$ is a 1-dimensional Cohen-Macaulay ring. To each point $p_{j}\in\operatorname{Supp}(\mathbb{X})$ we have the associated local ring ${\mathcal{O}}_{\mathbb{X},p_{j}}$. Its maximal ideal is denoted by $\mathfrak{m}_{\mathbb{X},p_{j}}$, and the residue field of $\mathbb{X}$ at $p_{j}$ is denoted by $\kappa(p_{j})$. The degree of $\mathbb{X}$ is defined as $\deg(\mathbb{X})=\sum_{j=1}^{s}\dim_{K}({\mathcal{O}}_{\mathbb{X},p_{j}})$.

Given any finitely generated graded $R_{\mathbb{X}}$-module $M$, the Hilbert function of $M$ is a map $\operatorname{HF}_{M}:\mathbb{Z}\rightarrow\mathbb{N}$ given by $\operatorname{HF}_{M}(i)=\dim_{K}(M_{i})$. The unique polynomial $\operatorname{HP}_{M}(z)\in\mathbb{Q}[z]$ for which $\operatorname{HF}_{M}(i)=\operatorname{HP}_{M}(i)$ for all $i\gg 0$ is called the Hilbert polynomial of $M$. The number

[TABLE]

is called the regularity index of $M$ (or of $\operatorname{HF}_{M}$). Whenever $\operatorname{HF}_{M}(i)=\operatorname{HP}_{M}(i)$ for all $i\in\mathbb{Z}$, we let $\operatorname{ri}(M)=-\infty.$ Instead of $\operatorname{HF}_{R_{\mathbb{X}}}$ we also write $\operatorname{HF}_{\mathbb{X}}$ and call it the Hilbert function of $\mathbb{X}$. Its regularity index is denoted by $r_{\mathbb{X}}.$ Note that $\operatorname{HF}_{\mathbb{X}}(i)=0$ for $i<0$ and

[TABLE]

and $\operatorname{HF}_{\mathbb{X}}(i)=\deg(\mathbb{X})$ for $i\geq r_{\mathbb{X}}$.

Definition 2.1**.**

Let $1\leq j\leq s$. A subscheme $\mathbb{X}^{\prime}\subsetneq\mathbb{X}$ is called a $p_{j}$-subscheme if the following conditions are satisfied:

1. (a)

${\mathcal{O}}_{\mathbb{X}^{\prime},p_{k}}={\mathcal{O}}_{\mathbb{X},p_{k}}$ for $k\neq j$. 2. (b)

The map ${\mathcal{O}}_{\mathbb{X},p_{j}}\twoheadrightarrow{\mathcal{O}}_{\mathbb{X}^{\prime},p_{j}}$ is an epimorphism.

A $p_{j}$-subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ is called maximal if $\deg(\mathbb{X}^{\prime})=\deg(\mathbb{X})-\dim_{K}\kappa(p_{j})$.

In case $\mathbb{X}$ has $K$-rational support (i.e., all points $p_{1},...,p_{s}$ are $K$-rational), a maximal $p_{j}$-subscheme of $\mathbb{X}$ is nothing but a subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ of degree $\deg(\mathbb{X}^{\prime})=\deg(\mathbb{X})-1$ with ${\mathcal{O}}_{\mathbb{X}^{\prime},p_{j}}\neq{\mathcal{O}}_{\mathbb{X},p_{j}}$. According to [13, Proposition 3.2], there is a 1-1 correspondence between a maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}$ and an ideal $\langle s_{j}\rangle$ in ${\mathcal{O}}_{\mathbb{X},p_{j}}$, where $s_{j}$ is an element in the socle $\operatorname{Ann}_{{\mathcal{O}}_{\mathbb{X},p_{j}}}(\mathfrak{m}_{\mathbb{X},p_{j}})$ of ${\mathcal{O}}_{\mathbb{X},p_{j}}$. The vanishing ideal of the scheme $\mathbb{X}^{\prime}$ in $R_{\mathbb{X}}$ is denoted by $I_{\mathbb{X}^{\prime}/\mathbb{X}}$ and its initial degree is given by $\alpha_{\mathbb{X}^{\prime}/\mathbb{X}}=\min\{\,i\in\mathbb{N}\mid(I_{\mathbb{X}^{\prime}/\mathbb{X}})_{i}\neq\langle 0\rangle\,\}$. We find a non-zero element $f_{\mathbb{X}}^{\prime}\in(I_{\mathbb{X}^{\prime}/\mathbb{X}})_{i}$, $i\geq\alpha_{\mathbb{X}^{\prime}/\mathbb{X}}$, such that $\tilde{\imath}(f_{\mathbb{X}}^{\prime})=(0,\dots,0,s_{j}T_{j}^{i},0,\dots,0)$, where the map

[TABLE]

is the injection given by $\tilde{\imath}(f)=(f_{p_{1}}T_{1}^{i},\dots,f_{p_{s}}T_{s}^{i})$, for $f\in(R_{\mathbb{X}})_{i}$ with $i\geq 0$, where $f_{p_{j}}$ is the germ of $f$ at $p_{j}$. Here the ring $Q^{h}(R_{\mathbb{X}})$ is the homogeneous ring of quotients of $R_{\mathbb{X}}$ defined as the localization of $R_{\mathbb{X}}$ with respect to the set of all homogeneous non-zerodivisors of $R_{\mathbb{X}}$ (cf. [13, Section 3]).

Let $\varkappa_{j}:=\dim_{K}\kappa(p_{j})$, and let $\{e_{j1},\dots,e_{j\varkappa_{j}}\}\subseteq{\mathcal{O}}_{\mathbb{X},p_{j}}$ be elements whose residue classes form a $K$-basis of $\kappa(p_{j})$. For $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$, we set

[TABLE]

Since the restriction map $\tilde{\imath}|_{(R_{\mathbb{X}})_{r_{\mathbb{X}}}}:(R_{\mathbb{X}})_{r_{\mathbb{X}}}\rightarrow(\prod_{j=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{j}}[T_{j},T_{j}^{-1}])_{r_{\mathbb{X}}}$ is an isomorphism of K-vector spaces, we have $\mu(a)\leq r_{\mathbb{X}}$ for all $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$. Using this notation, we recall from [14, Section 1] the following notion of separators.

Definition 2.2**.**

Let $\mathbb{X}^{\prime}$ be a maximal $p_{j}$-subscheme as above, and let

[TABLE]

and $f_{jk_{j}}=x_{0}^{r_{\mathbb{X}}-\mu(e_{jk_{j}}s_{j})}f^{*}_{jk_{j}}$ for $k_{j}=1,\dots,\varkappa_{j}$.

1. (a)

The set $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ is called the set of minimal separators of $\mathbb{X}^{\prime}$ in $\mathbb{X}$ with respect to $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$. 2. (b)

The set $\{f_{j1},\dots,f_{j\varkappa_{j}}\}$ is called the standard set of separators of $\mathbb{X}^{\prime}$ in $\mathbb{X}$ with respect to $s_{j}$ and $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$. 3. (c)

The number

[TABLE]

is called the maximal degree of a minimal separator of $\mathbb{X}^{\prime}$ in $\mathbb{X}$.

Remark 2.3**.**

Let $\mathbb{X}^{\prime}$ be a maximal $p_{j}$-subscheme of $\mathbb{X}$.

1. (a)

The maximal degree of a minimal separator of $\mathbb{X}^{\prime}$ in $\mathbb{X}$ depends neither on the choice of the socle element $s_{j}$ nor on the specific choice of $\{e_{j1},\dots,e_{j\varkappa_{j}}\}$ (see [13, Lemma 4.4]). Moreover, we have $\mu_{\mathbb{X}^{\prime}/\mathbb{X}}\leq r_{\mathbb{X}}$. 2. (b)

For $k_{j}=1,\dots,\varkappa_{j}$, let $F_{jk_{j}}$ (respectively, $F^{*}_{jk_{j}}$) be a representative of $f_{jk_{j}}$ (respectively, $f^{*}_{jk_{j}}$) in $P$. We also say that the set $\{F_{j1},\dots,F_{j\varkappa_{j}}\}$ is a standard set of separators of $\mathbb{X}^{\prime}$ in $\mathbb{X}$ and the set $\{F^{*}_{j1},\dots,F^{*}_{j\varkappa_{j}}\}$ is a set of minimal separators of $\mathbb{X}^{\prime}$ in $\mathbb{X}$. 3. (c)

According to [14, Proposition 2.5(c)], one may choose a set of minimal separators $\{f^{*}_{j1},\dots,f^{*}_{j\varkappa_{j}}\}$ of $\mathbb{X}^{\prime}$ in $\mathbb{X}$ such that

[TABLE]

for all $i\geq 0$.

Recall that a 0-dimensional scheme $\mathbb{X}$ is called a complete intersection if $I_{\mathbb{X}}$ can be generated by $n$ homogeneous polynomials in $P$, and it is called an arithmetically Gorenstein scheme if $R_{\mathbb{X}}$ is a Gorenstein ring. Note that every complete intersections are arithmetically Gorenstein, however, except for the case $n=2$, an arithmetically Gorenstein scheme is not a complete intersection in general (see [13, Example 2.12]).

In what follows, we let $\mathbb{W}\subseteq\mathbb{P}^{n}_{K}$ be a 0-dimensional arithmetically Gorenstein scheme, let $\mathbb{X}$ be a subscheme of $\mathbb{W}$, and let $I_{\mathbb{X}/\mathbb{W}}$ be the ideal of $\mathbb{X}$ in $R_{\mathbb{W}}$. Then the homogeneous ideal $\operatorname{Ann}_{R_{\mathbb{W}}}(I_{\mathbb{X}/\mathbb{W}})\subseteq R_{\mathbb{W}}$ is saturated and defines a 0-dimensional subscheme $\mathbb{Y}$ of $\mathbb{W}$.

Definition 2.4**.**

1. (a)

The subscheme $\mathbb{Y}\subseteq\mathbb{W}$ which is defined by the saturated homogeneous ideal $I_{\mathbb{Y}/\mathbb{W}}=\operatorname{Ann}_{R_{\mathbb{W}}}(I_{\mathbb{X}/\mathbb{W}})$ is said to be the residual scheme of $\mathbb{X}$ in $\mathbb{W}$. We also say that $\mathbb{X}$ and $\mathbb{Y}$ are (algebraically) linked by $\mathbb{W}$. 2. (b)

Two linked schemes $\mathbb{X}$ and $\mathbb{Y}$ by $\mathbb{W}$ are said to be geometrically linked by $\mathbb{W}$ if they have no common irreducible component.

Remark 2.5**.**

From the point of view of the saturated ideals, the schemes $\mathbb{X}$ and $\mathbb{Y}$ are geometrically linked by $\mathbb{W}$ if and only if $I_{\mathbb{W}}=I_{\mathbb{X}}\cap I_{\mathbb{Y}}$ and neither $I_{\mathbb{X}}$ nor $I_{\mathbb{Y}}$ is contained in any associated prime of the other (see [22, Section 5.2]). In this case, if we write $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$ and $\operatorname{Supp}(\mathbb{Y})=\{p^{\prime}_{1},\dots,p^{\prime}_{t}\}$, then we have $\operatorname{Supp}(\mathbb{W})=\{p_{1},\dots,p_{s},p^{\prime}_{1},\dots,p^{\prime}_{t}\}$ and $\operatorname{Supp}(\mathbb{X})\cap\operatorname{Supp}(\mathbb{Y})=\emptyset$. In particular, we have ${\mathcal{O}}_{\mathbb{W},p_{j}}={\mathcal{O}}_{\mathbb{X},p_{j}}$ for $j=1,\dots,s$ and ${\mathcal{O}}_{\mathbb{W},p^{\prime}_{j}}={\mathcal{O}}_{\mathbb{Y},p^{\prime}_{j}}$ for $j=1,\dots,t$.

First we collect some useful results about the linked schemes $\mathbb{X}$ and $\mathbb{Y}$ by the arithmetically Gorenstein scheme $\mathbb{W}$.

Proposition 2.6**.**

1. (a)

​We have $I_{\mathbb{X}/\mathbb{W}}=\operatorname{Ann}_{R_{\mathbb{W}}}(I_{\mathbb{Y}/\mathbb{W}})$. 2. (b)

We have $\deg(\mathbb{W})=\deg(\mathbb{X})+\deg(\mathbb{Y})$ and $r_{\mathbb{W}}=r_{\mathbb{X}}+\alpha_{\mathbb{Y}/\mathbb{W}}=r_{\mathbb{Y}}+\alpha_{\mathbb{X}/\mathbb{W}}$. 3. (c)

The Hilbert function of $I_{\mathbb{Y}/\mathbb{W}}$ satisfies

[TABLE]

Proof.

Claims (a) and (b) follow from [4]. To prove (c), we use (a) and (b) and [11, Proposition 2.2.9] to get the following sequence of isomorphism of graded $R_{\mathbb{W}}$-modules

[TABLE]

It is well known (cf. [15, Lemma 1.3]) that $\operatorname{HF}_{\operatorname{Hom}_{K[x_{0}]}(R_{\mathbb{X}},K[x_{0}])}(i)=\deg(\mathbb{X})-\operatorname{HF}_{\mathbb{X}}(-i-1)$ for all $i\in\mathbb{Z}$. Hence we get the desired formula for the Hilbert function of $I_{\mathbb{Y}/\mathbb{W}}$ and claim (c) follows. ∎

In the following we shall use “${}^{\overline{\;\;\,}}$” to denote residue classes modulo $X_{0}$.

Lemma 2.7**.**

For every $d\in\{1,\dots,r_{\mathbb{X}}\}$, we have $(\overline{I}_{\mathbb{W}})_{r_{\mathbb{W}}}:(\overline{I}_{\mathbb{Y}})_{\alpha_{\mathbb{Y}/\mathbb{W}}+(r_{\mathbb{X}}-d)}=(\overline{I}_{\mathbb{X}})_{d}$.

Proof.

Clearly, we have $I_{\mathbb{X}}\cdot I_{\mathbb{Y}}\subseteq I_{\mathbb{W}}$. This implies $(\overline{I}_{\mathbb{X}})_{d}\subseteq(\overline{I}_{\mathbb{W}})_{r_{\mathbb{W}}}\!:\!(\overline{I}_{\mathbb{Y}})_{\alpha_{\mathbb{Y}/\mathbb{W}}+(r_{\mathbb{X}}-d)}$. For the other inclusion, let $f\in(\overline{I}_{\mathbb{W}})_{r_{\mathbb{W}}}:(\overline{I}_{\mathbb{Y}})_{\alpha_{\mathbb{Y}/\mathbb{W}}+(r_{\mathbb{X}}-d)}$. In $\overline{R}_{\mathbb{W}}=R_{\mathbb{W}}/\langle x_{0}\rangle$, we have $\overline{f}\in(\operatorname{Ann}_{\overline{R}_{\mathbb{W}}}((\overline{I}_{\mathbb{Y}/\mathbb{W}})_{\alpha_{\mathbb{Y}/\mathbb{W}}+(r_{\mathbb{X}}-d)}))_{d}$. Since $\mathbb{W}$ is arithmetically Gorenstein, the ring $\overline{R}_{\mathbb{W}}$ is a [math]-dimensional local Gorenstein ring with socle $(\overline{R}_{\mathbb{W}})_{r_{\mathbb{W}}}\cong K$. Thus we can argue in the same way as Lemma 4.1 and Proposition 4.3.a of [7] to get

[TABLE]

Consequently, we have $\overline{f}\in(\overline{I}_{\mathbb{X}/\mathbb{W}})_{d}$, and hence $f\in(\overline{I}_{\mathbb{X}})_{d}$, as desired. ∎

The next lemma follows for instance from [20, 3.15 and 16.38-40].

Lemma 2.8**.**

Let $A/K$ be a finite Gorenstein algebra.

1. (a)

There is a non-degenerate $K$-bilinear form $\Phi:A\times A\rightarrow K$ with the property that $\Phi(xy,z)=\Phi(x,yz)$ for all $x,y,z\in A$. 2. (b)

Let $I$ be a non-zero ideal of $A$, and let $I^{0}=\{\,x\in A\,\mid\,\Phi(I,x)=0\,\}$. Then we have $\operatorname{Ann}_{A}(I)=I^{0}$ and $\dim_{K}I+\dim_{K}\operatorname{Ann}_{A}(I)=\dim_{K}A$.

A concrete description of the residual scheme in $\mathbb{W}$ of a maximal $p_{j}$-subscheme of $\mathbb{X}$ is given by the following proposition.

Proposition 2.9**.**

Let $\mathbb{W}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional arithmetically Gorenstein scheme, let $\mathbb{X}$ and $\mathbb{X}^{\prime}$ be subschemes of $\mathbb{W}$, let $\mathbb{Y}$ and $\mathbb{Y}^{\prime}$ be the residual schemes of $\mathbb{X}$ and $\mathbb{X}^{\prime}$ in $\mathbb{W}$ respectively, and let $p_{j}\in\operatorname{Supp}(\mathbb{X})$. Then $\mathbb{X}^{\prime}$ is a (maximal) $p_{j}$-subscheme of $\mathbb{X}$ if and only if $\mathbb{Y}^{\prime}$ contains $\mathbb{Y}$ as a (maximal) $p_{j}$-subscheme.

Proof.

As sets, we have $\operatorname{Supp}(\mathbb{W})=\operatorname{Supp}(\mathbb{X})\cup\operatorname{Supp}(\mathbb{Y})$ by [22, Proposition 5.2.2]. Let us write $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{t}\}$, $\operatorname{Supp}(\mathbb{Y})=\{p_{s+1},\dots,p_{u}\}$, and $\operatorname{Supp}(\mathbb{W})=\{p_{1},\dots,p_{s},p_{s+1},\dots,p_{t},p_{t+1},\dots,p_{u}\}$ with $s\leq t\leq u$. Then there exist ideals $\mathfrak{q}_{s+1}\subseteq{\mathcal{O}}_{\mathbb{W},p_{s+1}}$, $\dots$, $\mathfrak{q}_{t}\subseteq{\mathcal{O}}_{\mathbb{W},p_{t}}$ such that

[TABLE]

Consider the map $\theta:R_{\mathbb{W}}\rightarrow\prod_{j=1}^{u}{\mathcal{O}}_{\mathbb{W},p_{j}}$ given by $f\mapsto(f_{p_{1}},\dots,f_{p_{u}})$. According to [15, Lemma 1.1], the restriction $\theta|_{(R_{\mathbb{W}})_{i}}$ is an injection for $0\leq i<r_{\mathbb{W}}$ and is an isomorphism for all $i\geq r_{\mathbb{W}}$. By Proposition 2.6(b), we have $r_{\mathbb{X}}\leq r_{\mathbb{W}}$ and $\operatorname{HF}_{I_{\mathbb{X}/\mathbb{W}}}(i)=\deg(\mathbb{W})-\deg(\mathbb{X})=\deg(\mathbb{Y})$ for $i\geq r_{\mathbb{W}}$. Consequently, we get

[TABLE]

and $\dim_{K}\theta(I_{\mathbb{X}/\mathbb{W}})=\deg(\mathbb{Y})$. In $\prod_{j=1}^{u}{\mathcal{O}}_{\mathbb{W},p_{j}}$, we set

[TABLE]

It follows from Lemma 2.8 that $\deg(\mathbb{W})=\dim_{K}\Lambda+\dim_{K}\theta(I_{\mathbb{X}/\mathbb{W}})$. This implies $\dim_{K}\Lambda=\deg(\mathbb{W})-\deg(\mathbb{Y})$. Now we want to verify that $\theta(I_{\mathbb{Y}/\mathbb{W}})=\Lambda$. Since $\dim_{K}\theta(I_{\mathbb{Y}/\mathbb{W}})=\deg(\mathbb{W})-\deg(\mathbb{Y})=\dim_{K}\Lambda$, it suffices to show that $\Lambda\subseteq\theta(I_{\mathbb{Y}/\mathbb{W}})$. Let $i\geq 0$, let $f\in(R_{\mathbb{W}})_{i}\setminus\{0\}$ such that $\theta(f)\in\Lambda$, and let $g\in(I_{\mathbb{X}/\mathbb{W}})_{k}\setminus\{0\}$ with $k\geq\alpha_{\mathbb{X}/\mathbb{W}}$. Then we have $f\cdot g\in(R_{\mathbb{W}})_{i+k}$ and $\theta(f\cdot g)=\mathbf{0}$. This implies $f\cdot g=0$. Consequently, we have $f\in\operatorname{Ann}_{R_{\mathbb{W}}}(I_{\mathbb{X}/\mathbb{W}})=I_{\mathbb{Y}/\mathbb{W}}$, and hence $\Lambda\subseteq\theta(I_{\mathbb{Y}/\mathbb{W}})$.

Now we assume that $\mathbb{X}^{\prime}$ is a $p_{j}$-subscheme of $\mathbb{X}$. Then we have ${\mathcal{O}}_{\mathbb{X}^{\prime},p_{k}}={\mathcal{O}}_{\mathbb{X},p_{k}}$ for all $k\in\{1,\dots,u\}\setminus\{j\}$ and ${\mathcal{O}}_{\mathbb{X}^{\prime},p_{j}}={\mathcal{O}}_{\mathbb{W},p_{j}}/\mathfrak{q}^{\prime}_{j}$. We distinguish the following two cases.

Case (${\bf a}$) Suppose that $1\leq j\leq s$. We see that $\mathfrak{q}^{\prime}_{j}\neq\langle 0\rangle$ and

[TABLE]

This implies ${\mathcal{O}}_{\mathbb{Y}^{\prime},p_{k}}={\mathcal{O}}_{\mathbb{Y},p_{k}}$ for $k\neq j$ and ${\mathcal{O}}_{\mathbb{Y}^{\prime},p_{j}}={\mathcal{O}}_{\mathbb{W},p_{j}}/\operatorname{Ann}_{{\mathcal{O}}_{\mathbb{W},p_{j}}}(\mathfrak{q}^{\prime}_{j})\neq\langle 0\rangle={\mathcal{O}}_{\mathbb{Y},p_{j}}$. Hence $\mathbb{Y}$ is a $p_{j}$-subscheme of $\mathbb{Y}^{\prime}$.

Case (${\bf b}$) Suppose that $s+1\leq j\leq t$. We have $\mathfrak{q}^{\prime}_{j}\supsetneq\mathfrak{q}_{j}$ and

[TABLE]

This implies that ${\mathcal{O}}_{\mathbb{Y}^{\prime},p_{k}}={\mathcal{O}}_{\mathbb{Y},p_{k}}$ for $k\neq j$ and ${\mathcal{O}}_{\mathbb{Y}^{\prime},p_{j}}={\mathcal{O}}_{\mathbb{W},p_{j}}/\operatorname{Ann}_{{\mathcal{O}}_{\mathbb{W},p_{j}}}(\mathfrak{q}^{\prime}_{j})\neq{\mathcal{O}}_{\mathbb{W},p_{j}}/\operatorname{Ann}_{{\mathcal{O}}_{\mathbb{W},p_{j}}}(\mathfrak{q}_{j})={\mathcal{O}}_{\mathbb{Y},p_{j}}$. Again the scheme $\mathbb{Y}$ is a $p_{j}$-subscheme of $\mathbb{Y}^{\prime}$.

Conversely, if $\mathbb{Y}$ is a $p_{j}$-subscheme of $\mathbb{Y}^{\prime}$, where $p_{j}\in\operatorname{Supp}(\mathbb{X})$, an analogous argument as above yields that $\mathbb{X}^{\prime}$ is a $p_{j}$-subscheme of $\mathbb{X}$. ∎

3. Cayley-Bacharach Property and Liaison

In this section we use liaison techniques to characterize the Cayley-Bacharach property of a 0-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^{n}_{K}$. First we recall the notions of the degree of a point in $\mathbb{X}$ and the Cayley-Bacharach property (see [14, Section 4]).

Definition 3.1**.**

Let $d\geq 0$, let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme, and let $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$.

1. (a)

For $1\leq j\leq s$, the degree of $p_{j}$ in $\mathbb{X}$ is defined as

[TABLE]

where $\mu_{\mathbb{X}^{\prime}/\mathbb{X}}$ is the maximal degree of a minimal separator of $\mathbb{X}^{\prime}$ in $\mathbb{X}$. 2. (b)

We say that $\mathbb{X}$ has the Cayley-Bacharach property of degree $d$ (in short, $\mathbb{X}$ has CBP($d$)) if $\deg_{\mathbb{X}}(p_{j})\geq d+1$ for every $j\in\{1,\dots,s\}$. In the case that $\mathbb{X}$ has CBP($r_{\mathbb{X}}-1$) we also say that $\mathbb{X}$ is a Cayley-Bacharach scheme.

According to Remark 2.3(a), we have $0\leq\deg_{\mathbb{X}}(p_{j})\leq r_{\mathbb{X}}$. So, the number $r_{\mathbb{X}}-1$ is the largest degree $d\geq 0$ such that $\mathbb{X}$ can have CBP($d$). Hence it suffices to consider the Cayley-Bacharach property in degree $d\in\{0,\dots,r_{\mathbb{X}}-1\}$. Using standard sets of separators of $\mathbb{X}$, we can characterize the Cayley-Bacharach property as follows (see [14, Proposition 4.3]).

Proposition 3.2**.**

Let $0\leq d<r_{\mathbb{X}}$, let $\operatorname{Supp}(\mathbb{X})=\{p_{1},\dots,p_{s}\}$, and let $\varkappa_{j}=\dim\kappa(p_{j})$. Then the following statements are equivalent.

1. (a)

The scheme $\mathbb{X}$ has CBP($d$). 2. (b)

If $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ is a maximal $p_{j}$-subscheme and $\{f_{j1},\dots,f_{j\varkappa_{j}}\}\subseteq R_{\mathbb{X}}$ is a standard set of separators of $\mathbb{X}^{\prime}$ in $\mathbb{X}$, then there exists $k_{j}\in\{1\dots,\varkappa_{j}\}$ such that $x_{0}^{r_{\mathbb{X}}-d}\nmid f_{jk_{j}}$. 3. (c)

If $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ is a maximal $p_{j}$-subscheme and $\{F_{j1},\dots,F_{j\varkappa_{j}}\}\subseteq P$ is a standard set of separators of $\mathbb{X}^{\prime}$ in $\mathbb{X}$, then there exists $k_{j}\in\{1\dots,\varkappa_{j}\}$ such that $F_{jk_{j}}\notin\langle X_{0}^{r_{\mathbb{X}}-d},(I_{\mathbb{X}})_{r_{\mathbb{X}}}\rangle_{P}$. 4. (d)

For all $p_{j}\in\operatorname{Supp}(\mathbb{X})$, every maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ satisfies

[TABLE]

Now we give two useful lemmas that will be used in the proof of results below.

Lemma 3.3**.**

Let $\mathbb{W}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional arithmetically Gorenstein scheme, let $\mathbb{X}$ be a subscheme of $\mathbb{W}$ with its residual scheme $\mathbb{Y}$, and let $0\leq d<r_{\mathbb{X}}$. Furthermore, let $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ be a maximal $p_{j}$-subscheme, and let $\{F_{j1},\dots,F_{j\varkappa_{j}}\}\subseteq P_{r_{\mathbb{X}}}$ be a standard set of separators of $\mathbb{X}^{\prime}$ in $\mathbb{X}$. Suppose that $\langle F_{j1},\dots,F_{j\varkappa_{j}}\rangle_{K}\nsubseteq\langle X_{0}^{r_{\mathbb{X}}-d},(I_{\mathbb{X}})_{r_{\mathbb{X}}}\rangle_{P}$ and $\langle F_{j1},\dots,F_{j\varkappa_{j}}\rangle_{K}\subseteq\langle X_{0}^{r_{\mathbb{X}}-d-1},(I_{\mathbb{X}})_{r_{\mathbb{X}}}\rangle_{P}$, and write $F_{jk_{j}}=F^{\prime}_{jk_{j}}+X_{0}^{r_{\mathbb{X}}-d-1}G_{jk_{j}}$ with $F^{\prime}_{jk_{j}}\in(I_{\mathbb{X}})_{r_{\mathbb{X}}}$ and $G_{jk_{j}}\in P_{d+1}$.

Then there is $k_{j}\in\{1,\dots,\varkappa_{j}\}$ such that $G_{jk_{j}}\notin(I_{\mathbb{W}})_{r_{\mathbb{W}}}:(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$.

Proof.

Suppose that $G_{jk_{j}}\in(I_{\mathbb{W}})_{r_{\mathbb{W}}}:(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$ for all $k_{j}=1,\dots,\varkappa_{j}$. By modulo $X_{0}$ we have $\overline{G}_{jk_{j}}(\overline{I}_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}\subseteq(\overline{I}_{\mathbb{W}})_{r_{\mathbb{W}}}$. Note that $r_{\mathbb{W}}=\alpha_{\mathbb{Y}/\mathbb{W}}+r_{\mathbb{X}}$ by Proposition 2.6(b). Thus Lemma 2.7 yields that $\overline{G}_{jk_{j}}\in(\overline{I}_{\mathbb{X}})_{d+1}$. This allows us to write $G_{jk_{j}}=G^{\prime}_{jk_{j}}+X_{0}H_{jk_{j}}$ with $G^{\prime}_{jk_{j}}\in(I_{\mathbb{X}})_{d+1}$ and $H_{jk_{j}}\in P_{d}$. It is clear that $H_{jk_{j}}\in(I_{\mathbb{X}^{\prime}})_{d}$. From this we get $F_{jk_{j}}=(F^{\prime}_{jk_{j}}+X_{0}^{r_{\mathbb{X}}-d-1}G^{\prime}_{jk_{j}})+X_{0}^{r_{\mathbb{X}}-d}H_{jk_{j}}$ for all $k_{j}=1,\dots,\varkappa_{j}$. It follows that $F_{jk_{j}}\in\langle X_{0}^{r_{\mathbb{X}}-d},(I_{\mathbb{X}})_{r_{\mathbb{X}}}\rangle_{P}$ for all $k_{j}=1,\dots,\varkappa_{j}$. This is a contradiction to our hypothesis, and hence the claim is completely proved. ∎

Lemma 3.4**.**

Let $A$ be a 0-dimensional local affine $K$-algebra with maximal ideal $\mathfrak{m}$, let $\mathfrak{q}$ be a $\mathfrak{m}$-primary ideal, let $R=A/\mathfrak{q}$, and let $\pi:A\rightarrow R$ be the canonical epimorphism. Let $g\in A$ be an element such that $\pi(g)\in\operatorname{Ann}_{R}(\pi(\mathfrak{m}))$ is a non-zero socle element of $R$, and suppose $h\in\operatorname{Ann}_{A}(\mathfrak{q})$ and $gh\neq 0$.

1. (a)

We have $gh\in\operatorname{Ann}_{A}(\mathfrak{m})$ and $\langle 0\rangle:_{\langle g\rangle}\langle h\rangle\subseteq\mathfrak{q}$. 2. (b)

Every element $f\in A$ with $\pi(f)\in\langle\pi(g)\rangle_{R}\setminus\{0\}$ satisfies $fh\neq 0$. 3. (c)

Let $g_{1},...,g_{r}\in A\setminus\{0\}$. If the set $\{\pi(g_{1}),...,\pi(g_{r})\}\subseteq\langle\pi(g)\rangle_{R}$ is $K$-linearly independent, then the set $\{g_{1}h,...,g_{r}h\}$ is $K$-linearly independent.

Proof.

For (a), let $a\in\mathfrak{m}$ be a non-zero element. In $R$ we have $\pi(a)\in\pi(\mathfrak{m})$, and so we get $\pi(ag)=\pi(a)\pi(g)=0$ or $ag\in\mathfrak{q}$. It follows that $agh=0$. Hence $gh\in\operatorname{Ann}_{A}(\mathfrak{m})$. Moreover, for $f\in\langle 0\rangle:_{\langle g\rangle}\langle h\rangle$ we have $f=gf^{\prime}$ for some $f^{\prime}\in A$ and $gf^{\prime}h=fh=0$. Since $gh$ is a socle element of $A$ and $bgh\neq 0$ for $b\in A\setminus\mathfrak{m}$, we have $\operatorname{Ann}_{A}(gh)=\mathfrak{m}$. This implies $f^{\prime}\in\mathfrak{m}$. Thus $f=gf^{\prime}\in\mathfrak{q}$.

To prove (b), we consider an element $f\in A$ with $\pi(f)\in\langle\pi(g)\rangle_{R}\setminus\{0\}$. Writing $\pi(f)=\pi(g)\pi(f^{\prime})$ for some $f^{\prime}\in A\setminus\{0\}$, we see that $f^{\prime}\notin\mathfrak{m}$ is a unit and $f=gf^{\prime}+f^{\prime\prime}$ with $f^{\prime\prime}\in\mathfrak{q}$. So, we obtain $fh=f^{\prime}gh+f^{\prime\prime}h=f^{\prime}gh\neq 0$.

Next, we prove (c). Suppose that there are $a_{1},...,a_{r}\in K$ such that $a_{1}g_{1}h+\cdots+a_{r}g_{r}h=(a_{1}g_{1}+\cdots+a_{r}g_{r})h=0$. Since $\pi(a_{1}g_{1}+\cdots+a_{r}g_{r})\in\langle\pi(g)\rangle_{R}$, it follows from (b) that $\pi(a_{1}g_{1}+\cdots+a_{r}g_{r})=a_{1}\pi(g_{1})+\cdots+a_{r}\pi(g_{r})=0$. By assumption, we get $a_{1}=\cdots=a_{r}=0$. ∎

The first main result of this section is the following characterization of the Cayley-Bacharach property, which is a generalization of results for finite sets of $K$-rational points or for the case that $K$ is an algebraically closed field found in [7, Theorem 4.6] and [17, Theorem 4.1]. For $i\geq 0$ we write $F_{p}$ for the image in ${\mathcal{O}}_{\mathbb{W},p}$ of $F\in P_{i}$ under the composition map $P_{i}\rightarrow(R_{\mathbb{W}})_{i}\rightarrow\prod_{p\in\operatorname{Supp}(\mathbb{W})}{\mathcal{O}}_{\mathbb{W},p}\rightarrow{\mathcal{O}}_{\mathbb{W},p}$. Notice that $F\in I_{\mathbb{W}}$ if and only if $F_{p}=0$ for all $p\in\operatorname{Supp}(\mathbb{W})$ (cf. [15, Lemma 1.1]).

Theorem 3.5**.**

Let $\mathbb{W}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional arithmetically Gorenstein scheme, let $\mathbb{X}$ be a subscheme of $\mathbb{W}$, let $\mathbb{Y}$ be the residual scheme of $\mathbb{X}$ in $\mathbb{W}$, and let $0\leq d\leq r_{\mathbb{X}}-1$. Then the following statements are equivalent.

1. (a)

The scheme $\mathbb{X}$ has CBP($d$). 2. (b)

Every subscheme $\mathbb{Y}^{\prime}\subseteq\mathbb{W}$ containing $\mathbb{Y}$ as a maximal $p_{j}$-subscheme, where $p_{j}\in\operatorname{Supp}(\mathbb{X})$, satisfies $\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{Y}^{\prime}}}(r_{\mathbb{W}}-d-1)>0$. 3. (c)

We have $I_{\mathbb{W}}:(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}=I_{\mathbb{X}}$. 4. (d)

We have $(I_{\mathbb{W}})_{r_{\mathbb{W}}-1}:(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}=(I_{\mathbb{X}})_{d}$. 5. (e)

For all $p_{j}\in\operatorname{Supp}(\mathbb{X})$ and for every maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ with standard set of separators $\{F_{j1},...,F_{j\varkappa_{j}}\}$ there exists a homogeneous element $H_{j}\in(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$ such that $H_{j}\cdot\langle F_{j1},...,F_{j\varkappa_{j}}\rangle_{K}\nsubseteq I_{\mathbb{W}}$.

Proof.

First we prove the implication (a)$\Rightarrow$(b). Let $p_{j}\in\operatorname{Supp}(\mathbb{X})$, let $\varkappa_{j}=\dim_{K}\kappa(p_{j})$, let $\mathbb{Y}^{\prime}\subseteq\mathbb{W}$ be a subscheme containing $\mathbb{Y}$ as a maximal $p_{j}$-subscheme, and let $\mathbb{X}^{\prime}$ be the residual scheme of $\mathbb{Y}^{\prime}$ in $\mathbb{W}$. Proposition 2.9 shows that $\mathbb{X}^{\prime}$ is exactly a maximal $p_{j}$-subscheme of $\mathbb{X}$ of degree $\deg(\mathbb{X}^{\prime})=\deg(\mathbb{X})-\varkappa_{j}$. By Proposition 2.6, we observe that $r_{\mathbb{X}^{\prime}}+\alpha_{\mathbb{Y}^{\prime}/\mathbb{W}}=r_{\mathbb{W}}=r_{\mathbb{X}}+\alpha_{\mathbb{Y}/\mathbb{W}}$, and $\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{W}}}(i)=\deg(\mathbb{X})-\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{W}}-i-1)$ and $\operatorname{HF}_{I_{\mathbb{Y}^{\prime}/\mathbb{W}}}(i)=\deg(\mathbb{X}^{\prime})-\operatorname{HF}_{\mathbb{X}^{\prime}}(r_{\mathbb{W}}-i-1)$ for all $i\in\mathbb{Z}$. So, for all $i\in\mathbb{Z}$, we have $\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{Y}^{\prime}}}(i)=\varkappa_{j}-\operatorname{HF}_{I_{\mathbb{X}^{\prime}/\mathbb{X}}}(r_{\mathbb{W}}-i-1)$. According to Proposition 3.2, the Hilbert function of $I_{\mathbb{X}^{\prime}/\mathbb{X}}$ satisfies $\operatorname{HF}_{I_{\mathbb{X}^{\prime}/\mathbb{X}}}(d)<\varkappa_{j}$. Consequently, we get $\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{Y}^{\prime}}}(r_{\mathbb{W}}-d-1)=\varkappa_{j}-\operatorname{HF}_{I_{\mathbb{X}^{\prime}/\mathbb{X}}}(d)>0$, as wanted.

Now we prove the implication (b)$\Rightarrow$(c). Clearly, $I_{\mathbb{X}}\subseteq I_{\mathbb{W}}:(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$. Suppose for a contradiction that $F\in I_{\mathbb{W}}:(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$ and $F\notin I_{\mathbb{X}}$. There is a point $p_{j}\in\operatorname{Supp}(\mathbb{X})$ such that $F_{p_{j}}\neq 0$. By [19, Lemma 4.5.9(a)] there is $a_{j}\in{\mathcal{O}}_{\mathbb{X},p_{j}}$ such that $a_{j}\cdot F_{p_{j}}$ is a socle element of ${\mathcal{O}}_{\mathbb{X},p_{j}}$. This socle element defines a maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}$ of $\mathbb{X}$ by [13, Proposition 4.2]. Then the residual scheme $\mathbb{Y}^{\prime}$ of $\mathbb{X}^{\prime}$ in $\mathbb{W}$ satisfies $\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{Y}^{\prime}}}(r_{\mathbb{W}}-d-1)>0$ by Proposition 2.9 and (b). On the other hand, letting $G\in(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$, then $FG\in I_{\mathbb{W}}$ and $G\cdot I_{\mathbb{X}}\subseteq I_{\mathbb{W}}$. Since $I_{\mathbb{W}}$ is saturated, we have $G\cdot\langle F,I_{\mathbb{X}}\rangle^{\rm sat}\subseteq I_{\mathbb{W}}$. So, $G\cdot I_{\mathbb{X}^{\prime}}\subseteq I_{\mathbb{W}}$ or $G\in(I_{\mathbb{Y}^{\prime}})_{r_{\mathbb{W}}-d-1}$, as $I_{\mathbb{X}^{\prime}}\subseteq\langle F,I_{\mathbb{X}}\rangle^{\rm sat}$. Hence we get $\operatorname{HF}_{I_{\mathbb{Y}/\mathbb{Y}^{\prime}}}(r_{\mathbb{W}}-d-1)=0$, a contradiction.

Moreover, the implication (c)$\Rightarrow$(d) is clear. Next, we prove the implication (d)$\Rightarrow$(e). Let $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ be a maximal $p_{j}$-subscheme with set of minimal separators $\{F^{*}_{j1},\dots,F^{*}_{j\varkappa_{j}}\}$. If there exists some index $k_{j}\in\{1,\dots,\varkappa_{j}\}$ such that $\deg(F^{*}_{jk_{j}})\leq d$, then $G_{jk_{j}}=X_{0}^{d-\deg(F^{*}_{jk_{j}})}F^{*}_{jk_{j}}\notin(I_{\mathbb{X}})_{d}$, and so claim (d) implies $G_{jk_{j}}\notin(I_{\mathbb{W}})_{r_{\mathbb{W}}-1}:(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$. Let $H_{j}\in(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}\setminus\{0\}$ be such that $G_{jk_{j}}H_{j}\notin(I_{\mathbb{W}})_{r_{\mathbb{W}}-1}$. Since $X_{0}$ is a non-zerodivisor for $R_{\mathbb{W}}$, we have $F_{jk_{j}}H_{j}=X_{0}^{r_{\mathbb{X}}-d}G_{jk_{j}}H_{j}\notin I_{\mathbb{W}}$. In case $\deg(F^{*}_{jk_{j}})>d$ for all $k_{j}=1,\dots,\varkappa_{j}$, we see that $F_{jk_{j}}\notin\langle\,X_{0}^{r_{\mathbb{X}}-d},(I_{\mathbb{X}})_{r_{\mathbb{X}}}\,\rangle_{P}$. Let $1\leq\delta\leq r_{\mathbb{X}}-d$ be the smallest number such that $\langle F_{j1},\dots,F_{jk_{j}}\rangle_{K}\nsubseteq\langle X_{0}^{\delta},(I_{\mathbb{X}})_{r_{\mathbb{X}}}\rangle_{P}$. Write $F_{jk_{j}}=F^{\prime}_{jk_{j}}+X_{0}^{\delta-1}G_{jk_{j}}$ with $F^{\prime}_{jk_{j}}\in(I_{\mathbb{X}})_{r_{\mathbb{X}}}$ and $G_{jk_{j}}\in P_{r_{\mathbb{X}}-\delta+1}$. Then Lemma 3.3 yields that $G_{jk_{j}}\notin(I_{\mathbb{W}})_{r_{\mathbb{W}}}:(I_{\mathbb{Y}})_{\alpha_{\mathbb{Y}/\mathbb{W}}+\delta-1}$ for some $k_{j}\in\{1,\dots,\varkappa_{j}\}$. So, there is an element $\widetilde{H}_{j}\in(I_{\mathbb{Y}})_{\alpha_{\mathbb{Y}/\mathbb{W}}+\delta-1}$ such that $G_{jk_{j}}\widetilde{H}_{j}\notin(I_{\mathbb{W}})_{r_{\mathbb{W}}}$. Set $H_{j}=X_{0}^{r_{\mathbb{X}}-d-\delta}\widetilde{H}_{j}\in(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$. Since $F^{\prime}_{jk_{j}}\widetilde{H}_{j}\in I_{\mathbb{W}}$, we get $F_{jk_{j}}H_{j}\notin I_{\mathbb{W}}$.

Finally, we prove the implication (e)$\Rightarrow$(a). For a contradiction, assume that $\mathbb{X}$ does not have CBP($d$), and let $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ be a maximal $p_{j}$-subscheme such that its minimal separators satisfies $\deg(F^{*}_{jk_{j}})\leq d$ for all $k_{j}=1,\dots,\varkappa_{j}$. Set $G_{jk_{j}}=X_{0}^{d-\deg(F^{*}_{jk_{j}})}F^{*}_{jk_{j}}$ for $k_{j}=1,\dots,\varkappa_{j}$. By (e) there exists $H_{j}\in(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$ and some $k_{j}\in\{1,...,\varkappa_{j}\}$ such that $G_{jk_{j}}H_{j}\notin(I_{\mathbb{W}})_{r_{\mathbb{W}}-1}$. W.l.o.g. assume that $G_{j1}H_{j}\notin(I_{\mathbb{W}})_{r_{\mathbb{W}}-1}$. Notice that, as sets, $\operatorname{Supp}(\mathbb{W})=\operatorname{Supp}(\mathbb{X})\cup\operatorname{Supp}(\mathbb{Y})$. In ${\mathcal{O}}_{\mathbb{W},p_{j}}$, we have $(G_{j1}H_{j})_{p_{j}}\neq 0$ and $(G_{j1}H_{j})_{p}=0$ for any $p\in\operatorname{Supp}(\mathbb{W})\setminus\{p_{j}\}$. Also, by writing ${\mathcal{O}}_{\mathbb{X},p_{j}}={\mathcal{O}}_{\mathbb{W},p_{j}}/\mathfrak{q}_{j}$ for some ideal $\mathfrak{q}_{j}$ of ${\mathcal{O}}_{\mathbb{W},p_{j}}$, we have $\mathfrak{q}_{j}\cdot(H_{j})_{p_{j}}=\langle 0\rangle$ in ${\mathcal{O}}_{\mathbb{W},p}$ and $(G_{j1})_{p_{j}}\in{\mathcal{O}}_{\mathbb{X},p_{j}}$ is a socle element with $(G_{jk_{j}})_{p_{j}}\in\langle(G_{j1})_{p_{j}}\rangle_{{\mathcal{O}}_{\mathbb{X},p_{j}}}\setminus\{0\}$ for all $k_{j}=1,...,\varkappa_{j}$. In particular, by the definition of minimal separators, the set $\{(G_{j1})_{p_{j}},...,(G_{j\varkappa_{j}})_{p_{j}}\}$ is $K$-linearly independent. Thus Lemma 3.4 yields that $(G_{j1}H_{j})_{p_{j}}$ is a socle element of ${\mathcal{O}}_{\mathbb{W},p_{j}}$ and $\{(G_{j1}H_{j})_{p_{j}},\dots,(G_{j\varkappa_{j}}H_{j})_{p_{j}}\}\subseteq{\mathcal{O}}_{\mathbb{W},p_{j}}$ is $K$-linearly independent. Set $J:=\langle G_{jk}H_{j}+I_{\mathbb{W}}\mid 1\leq k\leq\varkappa_{j}\rangle_{R_{\mathbb{W}}}$. Obviously, we have

[TABLE]

for all $i\geq 0$. Furthermore, using [14, Lemma 2.8] we write

[TABLE]

for some $c_{j1l},\dots,c_{j\varkappa_{j}l}\in K$, where $0\leq i\leq n$ and $1\leq l\leq\varkappa_{j}$. Then we get

[TABLE]

and subsequently $\dim_{K}J_{r_{\mathbb{W}}-1+i}=\varkappa_{j}$ for all $i\geq 0$. Consequently, the homogeneous ideal $J$ defines a maximal $p_{j}$-subscheme $\mathbb{W}^{\prime}\subseteq\mathbb{W}$ such that $\dim_{K}(I_{\mathbb{W}^{\prime}/\mathbb{W}})_{r_{\mathbb{W}}-1}=\varkappa_{j}$. Therefore Proposition 3.2 implies that $\mathbb{W}$ is not a Cayley-Bacharach scheme. But $\mathbb{W}$ is arithmetically Gorenstein, and so it is a Cayley-Bacharach scheme by [13, Proposition 4.8], and this is a contradiction. ∎

Let us apply Theorem 3.5 to a concrete case.

Example 3.6**.**

Let $K$ be a field with ${\rm char}(K)\neq 2,3$, and let $\mathbb{W}\subseteq\mathbb{P}^{2}_{K}$ be the [math]-dimensional complete intersection defined by $I_{\mathbb{W}}=\langle F,G\rangle$, where $F=X_{1}(X_{1}-2X_{0})(X_{1}+2X_{0})$ and $G=(X_{2}-X_{0})(X_{1}^{2}+X_{2}^{2}-4X_{0}^{2})$. Then $\mathbb{W}$ has degree $9$ and the support of $\mathbb{W}$ is $\operatorname{Supp}(\mathbb{W})=\{p_{1},\dots,p_{7}\}$ with $p_{1}=(1:0:1)$, $p_{2}=(1:0:2)$, $p_{3}=(1:0:-2)$, $p_{4}=(1:2:1)$, $p_{5}=(1:2:0)$, $p_{6}=(1:-2:1)$, and $p_{7}=(1:-2:0)$. A homogeneous primary decomposition of $I_{\mathbb{W}}$ is $I_{\mathbb{W}}=I_{1}\cap\cdots\cap I_{7}$, where $I_{i}$ is the homogeneous prime ideal corresponding to $p_{i}$ for $i\neq 5,7$, $I_{5}=\langle X_{1}-2X_{0},X_{2}^{2}\rangle$, and $I_{7}=\langle X_{1}+2X_{0},X_{2}^{2}\rangle$. So, the scheme $\mathbb{W}$ is arithmetically Gorenstein, but not reduced at $p_{5}$ and $p_{7}$.

Now we consider the [math]-dimensional subscheme $\mathbb{X}$ of $\mathbb{W}$ defined by the ideal $I_{\mathbb{X}}=I_{1}\cap I_{3}\cap I_{4}\cap I_{5}\subseteq P$. Then $\deg(\mathbb{X})=5$ and $\mathbb{X}$ is not reduced. The residual scheme of $\mathbb{X}$ in $\mathbb{W}$ is denoted by $\mathbb{Y}$. It is easy to see that $\mathbb{X}$ and $\mathbb{Y}$ are geometrically linked. We have $r_{\mathbb{W}}=4$ and $r_{\mathbb{X}}=\alpha_{\mathbb{X}/\mathbb{W}}=r_{\mathbb{Y}}=\alpha_{\mathbb{Y}/\mathbb{W}}=2$. In this case there is a homogeneous polynomial $H\in(I_{\mathbb{Y}})_{2}$ such that its image in $R_{\mathbb{X}}$ is a non-zerodivisor, for instance, $H=X_{0}^{2}+X_{0}X_{1}+\tfrac{1}{4}X_{1}^{2}-\tfrac{1}{2}X_{0}X_{2}-\tfrac{1}{4}X_{1}X_{2}$. This polynomial satisfies the condition (e) in Theorem 3.5. Therefore $\mathbb{X}$ is a Cayley-Bacharach scheme.

The above example shows that, setting $I_{\mathbb{Y},\mathbb{X}}:=(I_{\mathbb{Y}}+I_{\mathbb{X}})/I_{\mathbb{X}}$, the condition $\operatorname{Ann}_{R_{\mathbb{X}}}((I_{\mathbb{Y},\mathbb{X}})_{r_{\mathbb{W}}-d-1})=\langle 0\rangle$ is a sufficient condition for $\mathbb{X}$ having CBP($d$) in this case. In general case, this is also true. Indeed, if $\operatorname{Ann}_{R_{\mathbb{X}}}((I_{\mathbb{Y},\mathbb{X}})_{r_{\mathbb{W}}-d-1})=\langle 0\rangle$ then for each maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ with standard set of separators $\{F_{j1},...,F_{j\varkappa_{j}}\}$ there is a non-zero homogeneous element $H_{j}\in(I_{\mathbb{Y},\mathbb{X}})_{r_{\mathbb{W}}-d-1}$ such that $(H_{j})_{p_{j}}\in{\mathcal{O}}_{\mathbb{X},p_{j}}\setminus\mathfrak{m}_{\mathbb{X},p_{j}}$, and so $(H_{j})_{p_{j}}\notin\mathfrak{m}_{\mathbb{W},p_{j}}$ and $(H_{j}F_{jk_{j}})_{p_{j}}\neq 0$ in ${\mathcal{O}}_{\mathbb{W},p_{j}}$. This means that $H_{j}F_{jk_{j}}\notin I_{\mathbb{W}}$. Subsequently, the condition (e) of Theorem 3.5 is satisfied, and hence $\mathbb{X}$ has CBP($d$).

However, the above condition is not a necessary condition for $\mathbb{X}$ having CBP($d$), as our next example shows.

Example 3.7**.**

Let $K$ be a field with ${\rm char}(K)\neq 2,3$, let $\mathbb{W}\subseteq\mathbb{P}^{2}_{K}$ be the [math]-dimensional complete intersection given in Example 3.6, and let $\mathbb{X}^{\prime}$ be the set of points in $\mathbb{W}$ with its homogeneous vanishing ideal $I_{\mathbb{X}^{\prime}}=I_{1}\cap I_{3}\cap I_{4}\cap I^{\prime}_{5}$, where $I^{\prime}_{5}=\langle X_{1}-2X_{0},X_{2}\rangle$ is the homogeneous prime ideal corresponding to $p_{5}$. Then the residual scheme $\mathbb{Y}^{\prime}$ of $\mathbb{X}^{\prime}$ in $\mathbb{W}$ has the homogeneous vanishing ideal $I_{\mathbb{Y}^{\prime}}=I_{2}\cap I^{\prime}_{5}\cap I_{6}\cap I_{7}$. It is clear that $r_{\mathbb{X}^{\prime}}=\alpha_{\mathbb{X}^{\prime}/\mathbb{W}}=r_{\mathbb{Y}^{\prime}}=\alpha_{\mathbb{Y}^{\prime}/\mathbb{W}}=2$ and

[TABLE]

In this case it is not difficult to verify that the scheme $\mathbb{X}^{\prime}$ is a complete intersection, and hence it is a Cayley-Bacharach scheme. However, there is no element $H$ in $(I_{\mathbb{Y}^{\prime}})_{2}$ such that $H_{p_{5}}\neq 0$ in ${\mathcal{O}}_{\mathbb{X}^{\prime},p_{5}}$. Hence the condition $\operatorname{Ann}_{R_{\mathbb{X}^{\prime}}}((I_{\mathbb{Y}^{\prime},\mathbb{X}^{\prime}})_{r_{\mathbb{W}}-r_{\mathbb{X}^{\prime}}})=\langle 0\rangle$ is not satisfied, even when $\mathbb{X}^{\prime}$ is a Cayley-Bacharach scheme. Moreover, we see that the element $F_{5}=X_{1}^{2}-2X_{1}X_{2}$ is a minimal separator of $\mathbb{X}^{\prime}\setminus\{p_{5}\}$ in $\mathbb{X}^{\prime}$ and $(F_{5}H_{5})_{p_{5}}$ is a socle element of ${\mathcal{O}}_{\mathbb{W},p_{5}}$, where $H_{5}=X_{0}^{2}-\tfrac{1}{4}X_{1}^{2}-\tfrac{1}{2}X_{0}X_{2}-\tfrac{1}{4}X_{1}X_{2}\in(I_{\mathbb{Y}^{\prime}})_{2}$.

It is interesting to examine the natural question whether the condition that $\mathbb{X}$ has CBP($d$) is equivalent to $\operatorname{Ann}_{R_{\mathbb{X}}}((I_{\mathbb{Y},\mathbb{X}})_{r_{\mathbb{W}}-d-1})=\langle 0\rangle$. When the schemes $\mathbb{W}$, $\mathbb{X}$ and $\mathbb{Y}$ are finite sets of $K$-rational points in $\mathbb{P}^{n}_{K}$ and $\mathbb{W}$ is a complete intersection, this question has an affirmative answer as was shown in [7, Theorem 4.6]. In our more general setting, this result can be generalized as follows.

Theorem 3.8**.**

​​​ Let $\mathbb{X}$​ and $\mathbb{Y}$​ be geometrically linked by a 0-dimensional arithmetically Gorenstein scheme $\mathbb{W}$, and let $I_{\mathbb{Y},\mathbb{X}}=(I_{\mathbb{Y}}+I_{\mathbb{X}})/I_{\mathbb{X}}$. Then the scheme $\mathbb{X}$ has CBP($d$) if and only if we have $\operatorname{Ann}_{R_{\mathbb{X}}}((I_{\mathbb{Y},\mathbb{X}})_{r_{\mathbb{W}}-d-1})=\langle 0\rangle$.

Proof.

According to the argument before Example 3.7, it suffices to show that $\operatorname{Ann}_{R_{\mathbb{X}}}((I_{\mathbb{Y},\mathbb{X}})_{r_{\mathbb{W}}-d-1})=\langle 0\rangle$ if $\mathbb{X}$ has CBP($d$). To this end, let $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ be a maximal $p_{j}$-subscheme with standard set of separators $\{F_{j1},\dots,F_{j\varkappa_{j}}\}\subseteq P_{r_{\mathbb{X}}}$. Since $\mathbb{X}$ has CBP($d$), Theorem 3.5 yields that there is an element $H_{j}\in(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$ such that $H_{j}\cdot\langle F_{j1},...,F_{j\varkappa_{j}}\rangle_{K}\nsubseteq I_{\mathbb{W}}$. W.l.o.g. we assume that $H_{j}F_{j1}\notin I_{\mathbb{W}}$. Since $\mathbb{X}$ and $\mathbb{Y}$ are geometrically linked, $(F_{j1})_{p_{j}}$ is a socle element in ${\mathcal{O}}_{\mathbb{W},p_{j}}={\mathcal{O}}_{\mathbb{X},p_{j}}$. Since $(H_{j}F_{j1})_{p}=0$ in ${\mathcal{O}}_{\mathbb{W},p}$ for every $p\in\operatorname{Supp}(\mathbb{W})\setminus\{p_{j}\}$ and $(H_{j}F_{j1})_{p_{j}}\neq 0$, we get $(H_{j})_{p_{j}}\notin\mathfrak{m}_{\mathbb{X},p_{j}}$. Consequently, for each point $p_{j}$ of $\operatorname{Supp}(\mathbb{X})$, we can find an element $H_{j}\in(I_{\mathbb{Y}})_{r_{\mathbb{W}}-d-1}$ such that $(H_{j})_{p_{j}}$ is a unit of ${\mathcal{O}}_{\mathbb{X},p_{j}}$. By [15, Lemma 1.1], this condition is exactly the right condition to have $\operatorname{Ann}_{R_{\mathbb{X}}}((I_{\mathbb{Y},\mathbb{X}})_{r_{\mathbb{W}}-d-1})=\langle 0\rangle$. ∎

Remark 3.9**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a [math]-dimensional scheme.

1. (a)

If $\mathbb{X}$ is reduced and has $K$-rational support, then there is a complete intersection consisting of distinct $K$-rational points $\mathbb{W}$ containing $\mathbb{X}$ such that $\mathbb{X}$ and its residue scheme by $\mathbb{W}$ are geometrically linked (see, e.g., [7, Remark 4.11]). 2. (b)

If ${\mathcal{O}}_{\mathbb{X},p_{j}}$ is not a Gorenstein local ring for some point $p_{j}\in\operatorname{Supp}(\mathbb{X})$, then there is no [math]-dimensional arithmetically Gorenstein scheme $\mathbb{W}\subseteq\mathbb{P}^{n}_{K}$ containing $\mathbb{X}$ such that $\mathbb{X}$ and its residual scheme in $\mathbb{W}$ are geometrically linked.

We end this section with the following immediate consequence of Theorem 3.5. This result allows us to check whether $\mathbb{X}$ has CBP($d$) by using a truncated Gröbner basis calculation (cf. [18, Section 4.5]). For the case of sets of distinct $K$-rational points and $d=r_{\mathbb{X}}-1$ see also [7, Corollary 4.10].

Corollary 3.10**.**

In the setting of Theorem 3.5, the scheme $\mathbb{X}$ has CBP($d$) if and only if $\operatorname{HF}_{P/(I_{\mathbb{W}}:(I_{\mathbb{W}}:I_{\mathbb{X}})_{r_{\mathbb{W}}-d-1})}(d)=\operatorname{HF}_{\mathbb{X}}(d)$.

4. Bound the Hilbert Function of the Dedekind Different

In this section, we let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a 0-dimensional scheme and we let $0\leq d<r_{\mathbb{X}}$. The aim of this section is to characterize the Cayley-Bacharach property using the canonical module of $R_{\mathbb{X}}$, and apply these results to bound the Hilbert function and determine the regularity index of the Dedekind different of $\mathbb{X}$ under some additional hypotheses.

Recall that the graded $R_{\mathbb{X}}$-module $\omega_{R_{\mathbb{X}}}=\underline{\operatorname{Hom}}_{K[x_{0}]}(R_{\mathbb{X}},K[x_{0}])(-1)$ is called the canonical module of $R_{\mathbb{X}}$. Its $R_{\mathbb{X}}$-module structure is defined by $(f\cdot\varphi)(g)=\varphi(fg)$ for all $f,g\in R_{\mathbb{X}}$ and $\varphi\in\omega_{R_{\mathbb{X}}}$. It is also a finitely generated graded $R_{\mathbb{X}}$-module and

[TABLE]

The following two lemmas give us some more information about this module.

Lemma 4.1**.**

For every homogeneous element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ its restriction $\overline{\varphi}=\varphi|_{(R_{\mathbb{X}})_{d+1}}:(R_{\mathbb{X}})_{d+1}\rightarrow K$ is a $K$-linear map such that $\overline{\varphi}(x_{0}(R_{\mathbb{X}})_{d})\!=\!\langle 0\rangle$. Conversely, if $\overline{\varphi}:(R_{\mathbb{X}})_{d+1}\rightarrow K$ is a $K$-linear map such that $\overline{\varphi}(x_{0}(R_{\mathbb{X}})_{d})=\langle 0\rangle$, then there exists a homogeneous element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ such that $\varphi|_{(R_{\mathbb{X}})_{d+1}}=\overline{\varphi}$.

Proof.

Clearly, for every homogeneous element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ its restriction $\overline{\varphi}=\varphi|_{(R_{\mathbb{X}})_{d+1}}$ is a $K$-linear map. Also, we have

[TABLE]

Now let $\overline{\varphi}:(R_{\mathbb{X}})_{d+1}\rightarrow K$ is a $K$-linear map such that $\overline{\varphi}(x_{0}(R_{\mathbb{X}})_{d})=\langle 0\rangle$. Let $h_{i}=\operatorname{HF}_{\mathbb{X}}(i)-\operatorname{HF}_{\mathbb{X}}(i-1)$ for $i\in\mathbb{N}$. Note that $(R_{\mathbb{X}})_{i}=x_{0}^{i-r_{\mathbb{X}}}(R_{\mathbb{X}})_{r_{\mathbb{X}}}$ and $h_{i}=0$ for all $i>r_{\mathbb{X}}$. To define an element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ with the desired properties, we start taking a $K$-basis $g_{1},...,g_{\scriptscriptstyle\sum\limits_{0\leq k\leq d+1}h_{k}}$ of $(R_{\mathbb{X}})_{d+1}$. For $i=d+2,...,r_{\mathbb{X}}$, we choose $g_{\scriptscriptstyle\sum\limits_{0\leq k<i}h_{k}+1},...,g_{\scriptscriptstyle\sum\limits_{0\leq k\leq i}h_{k}}$ such that the set

[TABLE]

forms a $K$-basis of $(R_{\mathbb{X}})_{i}$. Then we get

[TABLE]

for all $i\geq r_{\mathbb{X}}$. Let $\varphi:R_{\mathbb{X}}\rightarrow K[x_{0}]$ be the homogeneous $K$-linear map of degree $-d$ defined as: for $f\in R_{i}$ with $i\leq d$ we let $\varphi(f)=0$, and for $f\in R_{i}$ with $i\geq d+1$ we write

[TABLE]

and let $\varphi(f)=\sum_{1\leq j\leq\scriptscriptstyle\sum\limits_{0\leq k\leq d+1}h_{k}}a_{j}x_{0}^{i-d-1}\overline{\varphi}(g_{j})$. The condition $\overline{\varphi}(x_{0}(R_{\mathbb{X}})_{d})=\langle 0\rangle$ implies that the map $\varphi$ is $K[x_{0}]$-linear. Hence $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ is the desired element that we wanted to construct. ∎

Lemma 4.2**.**

The canonical module $\omega_{R_{\mathbb{X}}}$ satisfies $\operatorname{Ann}_{R_{\mathbb{X}}}((\omega_{R_{\mathbb{X}}})_{-d})=\langle 0\rangle$ if and only if for every $p_{j}\in\operatorname{Supp}(\mathbb{X})$ and for every maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ there exists a homogeneous element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ such that $I_{\mathbb{X}^{\prime}/\mathbb{X}}\cdot\varphi\neq\langle 0\rangle$.

Proof.

We need only to prove that if for every $p_{j}\in\operatorname{Supp}(\mathbb{X})$ and for every maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ there exists a homogeneous element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ such that $I_{\mathbb{X}^{\prime}/\mathbb{X}}\cdot\varphi\neq\langle 0\rangle$ then $\operatorname{Ann}_{R_{\mathbb{X}}}((\omega_{R_{\mathbb{X}}})_{-d})=\langle 0\rangle$. Suppose for a contradiction that $f\cdot(\omega_{R_{\mathbb{X}}})_{-d}=\langle 0\rangle$ for some $f\in(R_{\mathbb{X}})_{i}\setminus\{0\}$ with $i\geq 0$. Since $f\neq 0$, we may assume the germ $f_{p_{j}}\neq 0$ for some $j\in\{1,...,s\}$. In the local ring ${\mathcal{O}}_{\mathbb{X},p_{j}}$ we find an element $a\in{\mathcal{O}}_{\mathbb{X},p_{j}}$ such that $s_{j}=af_{p_{j}}$ is a socle element of ${\mathcal{O}}_{\mathbb{X},p_{j}}$ (cf. [19, Lemma 4.5.9(a)]). Now let $g=\tilde{\imath}^{-1}((0,...,0,s_{j}T_{j}^{r_{\mathbb{X}}},0,...,0))$ and $h=\tilde{\imath}^{-1}((0,...,0,aT_{j}^{r_{\mathbb{X}}},0,...,0))$. Then $g,h\in(R_{\mathbb{X}})_{r_{\mathbb{X}}}$ satisfies $x_{0}^{i}g=fh$. Also, the ideal $\langle g\rangle$ defines a maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}$ of $\mathbb{X}$, that is, we have $I_{\mathbb{X}^{\prime}/\mathbb{X}}=\langle g\rangle^{\rm sat}$. Thus there is $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ such that $\langle g\rangle^{\rm sat}\cdot\varphi\neq\langle 0\rangle$, in particularly, $g\cdot\varphi\neq 0$. It follow that $g\cdot\varphi(\widetilde{g})\neq 0$ for some non-zero homogeneous element $\widetilde{g}\in R_{\mathbb{X}}$. Hence we get $0=(f\cdot\varphi)(h\widetilde{g})=(fh\cdot\varphi)(\widetilde{g})=(x_{0}^{i}g\cdot\varphi)(\widetilde{g})=(g\cdot\varphi)(x_{0}^{i}\widetilde{g})=x_{0}^{i}(g\cdot\varphi)(\widetilde{g})\neq 0$, a contradiction. ∎

Using the above properties we prove the following characterization of the Cayley-Bacharach property in terms of the canonical module.

Proposition 4.3**.**

Let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a 0-dimensional scheme, and let $0\leq d<r_{\mathbb{X}}$. Then the following conditions are equivalent.

1. (a)

The scheme $\mathbb{X}$ has CBP($d$). 2. (b)

We have $\operatorname{Ann}_{R_{\mathbb{X}}}((\omega_{R_{\mathbb{X}}})_{-d})=\langle 0\rangle$.

Proof.

Suppose that $\mathbb{X}$ has CBP($d$). Let $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ be a maximal $p_{j}$-subscheme with set of minimal separators $\{f^{*}_{j1},...,f^{*}_{j\varkappa_{j}}\}$. By Proposition 3.2, there exists an index $k\in\{1,...,\varkappa_{j}\}$ such that $\rho=\deg(f^{*}_{jk})\geq d+1$ and $f^{*}_{jk}\notin x_{0}(R_{\mathbb{X}})_{\rho-1}$. W.l.o.g. assume that $k=1$. So, we can define a $K$-linear map $\overline{\varphi}_{j}:(R_{\mathbb{X}})_{\rho}\rightarrow K$ such that $\overline{\varphi}_{j}(x_{0}(R_{\mathbb{X}})_{\rho-1})=\langle 0\rangle$ and $\overline{\varphi}_{j}(f^{*}_{j1})\neq 0$. Using Lemma 4.1 we lift this map to obtain a homogeneous element $\varphi_{j}\in(\omega_{R_{\mathbb{X}}})_{-\rho+1}$ such that $\varphi_{j}(f^{*}_{j1})\neq 0$. Since $x_{0}$ is a non-zerodivisor of $R_{\mathbb{X}}$, it follows that $x_{0}^{\rho-d-1}\varphi_{j}(f^{*}_{j1})\neq 0$. Especially, we have $x_{0}^{\rho-d-1}\cdot\varphi_{j}\in(\omega_{R_{\mathbb{X}}})_{-d}$ and $I_{\mathbb{X}^{\prime}/\mathbb{X}}\cdot(x_{0}^{\rho-d-1}\cdot\varphi_{j})\neq\langle 0\rangle$. Hence Lemma 4.2 yields the condition $\operatorname{Ann}_{R_{\mathbb{X}}}((\omega_{R_{\mathbb{X}}})_{-d})=\langle 0\rangle$.

Conversely, assume for a contradiction that $\mathbb{X}$ does not have CBP($d$). There is a maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}\subseteq\mathbb{X}$ such that its set of minimal separators $\{f^{*}_{j1},...,f^{*}_{j\varkappa_{j}}\}$ satisfies $\deg(f^{*}_{jk})\leq d$ for all $k=1,...,\varkappa_{j}$. By Remark 2.3(a)-(c), we may assume that, for $i\geq 0$, the set $\{x_{0}^{i-\deg(f^{*}_{jk_{j}})}f^{*}_{jk_{j}}\,\mid\,\deg(f^{*}_{jk_{j}})\leq i\}$ is a $K$-basis of $(I_{\mathbb{X}^{\prime}/\mathbb{X}})_{i}$. In this case for every $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ we have $\varphi(f^{*}_{jk})=0$ for all $k=1,...,\varkappa_{j}$. We shall show that $f^{*}_{jk}\cdot\varphi=0$ for all $k=1,...,\varkappa_{j}$. Let $i\geq 0$ and let $h\in R_{i}\setminus\{0\}$ be a homogeneous element. If $hf^{*}_{jk}=0$ then $(f^{*}_{jk}\cdot\varphi)(h)=\varphi(hf^{*}_{jk})=0$. Suppose that $hf^{*}_{jk}\neq 0$. Since $hf^{*}_{jk}\in I_{\mathbb{X}^{\prime}/\mathbb{X}}$, this allows us to write $hf^{*}_{jk}=\sum_{l=1}^{\varkappa_{j}}c_{jl}x_{0}^{i+\deg(f^{*}_{jk})-\deg(f^{*}_{jl})}f^{*}_{jl}$ for some $c_{j1},...,c_{j\varkappa_{j}}\in K$. This implies $(f^{*}_{jk}\cdot\varphi)(h)=\varphi(hf^{*}_{jk})=\sum_{l=1}^{\varkappa_{j}}c_{jl}x_{0}^{i+\deg(f^{*}_{jk})-\deg(f^{*}_{jl})}\varphi(f^{*}_{jl})=0$. Hence we have shown $f^{*}_{jk}\cdot\varphi=0$ for all $k=1,...,\varkappa_{j}$. In addition, we have $I_{\mathbb{X}^{\prime}/\mathbb{X}}=\langle f^{*}_{j1},...,f^{*}_{j\varkappa_{j}}\rangle$. It follows that $I_{\mathbb{X}^{\prime}/\mathbb{X}}\cdot\varphi=\langle 0\rangle$ for any homogeneous element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$. Therefore we get $\operatorname{Ann}_{R_{\mathbb{X}}}((\omega_{R_{\mathbb{X}}})_{-d})\neq\langle 0\rangle$, a contradiction. ∎

As a consequence of the proposition, we get the following property. Here we recall that a 0-dimensional scheme $\mathbb{X}$ is called locally Gorenstein if the local ring ${\mathcal{O}}_{\mathbb{X},p_{j}}$ is a Gorenstein ring for every point $p_{j}\in\operatorname{Supp}(\mathbb{X})$.

Corollary 4.4**.**

Let $K$ be an infinite field, let $\mathbb{X}\subseteq\mathbb{P}^{n}_{K}$ be a 0-dimensional locally Gorenstein scheme, and let $0\leq d<r_{\mathbb{X}}$. Then $\mathbb{X}$ has CBP($d$) if and only if there exists an element $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ such that $\operatorname{Ann}_{R_{\mathbb{X}}}(\varphi)=\langle 0\rangle$.

Proof.

Since $\mathbb{X}$ is locally Gorenstein, there is for each point $p_{j}\in\operatorname{Supp}(\mathbb{X})$ a uniquely maximal $p_{j}$-subscheme $\mathbb{X}^{\prime}_{j}$ of $\mathbb{X}$. So, the condition (b) of Proposition 4.3 is equivalent to the condition that for each $j\in\{1,...,s\}$ there exists $\varphi_{j}\in(\omega_{R_{\mathbb{X}}})_{-d}$ such that $I_{\mathbb{X}^{\prime}_{j}/\mathbb{X}}\cdot\varphi_{j}\neq\langle 0\rangle$. This is in turn equivalent to that there exists $\varphi\in(\omega_{R_{\mathbb{X}}})_{-d}$ such that $I_{\mathbb{X}^{\prime}_{j}/\mathbb{X}}\cdot\varphi\neq\langle 0\rangle$ for $j=1,...,s$, since the base field $K$ is infinite, and this condition is exactly the right condition to make $\operatorname{Ann}_{R_{\mathbb{X}}}(\varphi)=\langle 0\rangle$. ∎

Remark 4.5**.**

This corollary is a generalization of a result for the case $d=r_{\mathbb{X}}-1$ found in [13, Proposition 4.12]. Moreover, the hypothesis in the corollary that $K$ is infinite is necessary (cf. [13, Example 4.14]).

Now let us apply the above results to look at the Hilbert function of the Dedekind different of $\mathbb{X}$. For this purpose, we assume, in what follows, that $\mathbb{X}$ is locally Gorenstein, and we let $L_{0}=K[x_{0},x_{0}^{-1}]$. The homogeneous ring of quotients of $R_{\mathbb{X}}$ is $Q^{h}(R_{\mathbb{X}})\cong\prod_{j=1}^{s}{\mathcal{O}}_{\mathbb{X},p_{j}}[T_{j},T_{j}^{-1}]$. According to [13, Proposition 3.3], the graded algebra $Q^{h}(R_{\mathbb{X}})/L_{0}$ has a homogeneous trace map $\sigma$ of degree zero, i.e., $\sigma\in(\underline{\operatorname{Hom}}_{L_{0}}(Q^{h}(R_{\mathbb{X}}),L_{0}))_{0}$ satisfies $\underline{\operatorname{Hom}}_{L_{0}}(Q^{h}(R_{\mathbb{X}}),L_{0})=Q^{h}(R_{\mathbb{X}})\cdot\sigma$. Thus there is an injective homomorphism of graded $R_{\mathbb{X}}$-modules

[TABLE]

The image of $\Phi$ is a homogeneous fractional $R_{\mathbb{X}}$-ideal $\mathfrak{C}^{\sigma}_{\mathbb{X}}$ of $Q^{h}(R_{\mathbb{X}})$. It is also a finitely generated graded $R_{\mathbb{X}}$-module and

[TABLE]

Definition 4.6**.**

The $R$-module $\mathfrak{C}^{\sigma}_{\mathbb{X}}$ is called the Dedekind complementary module of $\mathbb{X}$ with respect to $\sigma$. Its inverse,

[TABLE]

is called the Dedekind different of $\mathbb{X}$ with respect to $\sigma$.

The following basic properties of the Dedekind different of $\mathbb{X}$ are shown in [13, Proposition 3.7].

Proposition 4.7**.**

Let $\sigma$ be a trace map of $Q^{h}(R_{\mathbb{X}})/L_{0}$.

1. (a)

The Dedekind different $\delta^{\sigma}_{\mathbb{X}}$ is a homogeneous ideal of $R_{\mathbb{X}}$ and $x_{0}^{2r_{\mathbb{X}}}\in\delta^{\sigma}_{\mathbb{X}}$. 2. (b)

The Hilbert function of $\delta^{\sigma}_{\mathbb{X}}$ satisfies $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)=0$ for $i<0$, $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)=\deg(\mathbb{X})$ for $i\geq 2r_{\mathbb{X}}$, and $0\leq\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(0)\leq\cdots\leq\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(2r_{\mathbb{X}})=\deg(\mathbb{X})$. In particular, the regularity index of $\delta^{\sigma}_{\mathbb{X}}$ satisfies $r_{\mathbb{X}}\leq\operatorname{ri}(\delta^{\sigma}_{\mathbb{X}})\leq 2r_{\mathbb{X}}$.

When $\mathbb{X}$ has CBP($d$), the Hilbert function of the Dedekind different and its regularity index can be described as follows. We use the notation $\alpha_{\delta}=\min\{i\in\mathbb{N}\mid(\delta^{\sigma}_{\mathbb{X}})_{i}\neq\langle 0\rangle\}$.

Proposition 4.8**.**

Let $K$ be an infinite field, let $\sigma$ be a trace map of $Q^{h}(R_{\mathbb{X}})/L_{0}$, and suppose that $\mathbb{X}$ has CBP($d$) with $0\leq d\leq r_{\mathbb{X}}-1$.

1. (a)

We have $d+1\leq\alpha_{\delta}\leq 2r_{\mathbb{X}}$ and $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)\leq\operatorname{HF}_{\mathbb{X}}(i-d-1)$ for all $i\in\mathbb{Z}$. 2. (b)

Let $i_{0}$ be the smallest number such that $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i_{0})=\operatorname{HF}_{\mathbb{X}}(i_{0}-d-1)>0$. Then we have $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)=\operatorname{HF}_{\mathbb{X}}(i-d-1)$ for all $i\geq i_{0}$ and

[TABLE]

Proof.

Since $\mathfrak{C}_{\mathbb{X}}^{\sigma}\cong\omega_{R_{\mathbb{X}}}(1)$, Corollary 4.4 implies that there is $g\in(\mathfrak{C}_{\mathbb{X}}^{\sigma})_{-d-1}$ such that $\operatorname{Ann}_{R_{\mathbb{X}}}(g)=\langle 0\rangle$. Notice that $x_{0}$ is a non-zerodivisor of $R_{\mathbb{X}}$. Then we find a non-zerodivisor $\widetilde{g}\in(R_{\mathbb{X}})_{r_{\mathbb{X}}}$ such that $g=x_{0}^{-r_{\mathbb{X}}-d-1}\widetilde{g}$ by [13, Proposition 3.7]. Observe that $\widetilde{g}\cdot(\delta^{\sigma}_{\mathbb{X}})_{i}\subseteq x_{0}^{r_{\mathbb{X}}+d+1}(R_{\mathbb{X}})_{i-d-1}$. This implies $(\delta^{\sigma}_{\mathbb{X}})_{i}=\langle 0\rangle$ for $i\leq d$, and so $d+1\leq\alpha_{\delta}$. Moreover, for all $i\in\mathbb{Z}$, we have

[TABLE]

Thus claim (a) is completely proved.

Now we prove claim (b). Clearly, we have $d+1\leq i_{0}\leq 2r_{\mathbb{X}}$. By induction, we only need to show that $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i_{0}+1)=\operatorname{HF}_{\mathbb{X}}(i_{0}-d)>0$. Let $f\in(R_{\mathbb{X}})_{i_{0}-d}\setminus\{0\}$. There are $g_{0},\dots,g_{n}\in(R_{\mathbb{X}})_{i_{0}-d-1}$ such that $f=x_{0}g_{0}+x_{1}g_{1}+\cdots+x_{n}g_{n}$. By assumption, we have $\widetilde{g}\cdot(\delta^{\sigma}_{\mathbb{X}})_{i_{0}}=x_{0}^{r_{\mathbb{X}}+d+1}(R_{\mathbb{X}})_{i_{0}-d-1}$. This enables us to write $x_{0}^{r_{\mathbb{X}}+d+1}g_{j}=\widetilde{g}h_{j}$ for some $h_{j}\in(\delta^{\sigma}_{\mathbb{X}})_{i_{0}}$, where $j\in\{0,\dots,n\}$. Thus we have

[TABLE]

and so $x_{0}^{r_{\mathbb{X}}+d+1}f\in\widetilde{g}\cdot(\delta^{\sigma}_{\mathbb{X}})_{i_{0}+1}$. Hence $x_{0}^{r_{\mathbb{X}}+d+1}(R_{\mathbb{X}})_{i_{0}-d}\!=\!\widetilde{g}\cdot(\delta^{\sigma}_{\mathbb{X}})_{i_{0}+1}$. In other words, we get $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i_{0}+1)=\operatorname{HF}_{\mathbb{X}}(i_{0}-d)$.

Let k=\max\big{\{}\,i_{0},r_{\mathbb{X}}+d+1\,\big{\}}. In order to prove the equality $\operatorname{ri}(\delta^{\sigma}_{\mathbb{X}})=k$, we consider the following two cases.

Case (1) Let $i_{0}\geq r_{\mathbb{X}}+d+1$. Then we have $k=i_{0}$. Observe that

[TABLE]

It follows that $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(k)=\deg(\mathbb{X})$, and hence $k\geq\operatorname{ri}(\delta^{\sigma}_{\mathbb{X}})$. Moreover, for $i<k=i_{0}$, we have $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)<\operatorname{HF}_{\mathbb{X}}(i-d-1)\leq\operatorname{HF}_{\mathbb{X}}(k-d-1)=\deg(\mathbb{X})$. Thus we get $\operatorname{ri}(\delta^{\sigma}_{\mathbb{X}})=k$.

Case (2) Let $i_{0}<r_{\mathbb{X}}+d+1$. Then we have $k=r_{\mathbb{X}}+d+1$ and $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(k)=\operatorname{HF}_{\mathbb{X}}(k-d-1)=\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}})=\deg(\mathbb{X})$. This implies $k\geq\operatorname{ri}(\delta^{\sigma}_{\mathbb{X}})$. For $i<k$, we have $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)\leq\operatorname{HF}_{\mathbb{X}}(i-d-1)\leq\operatorname{HF}_{\mathbb{X}}(r_{\mathbb{X}}-1)<\deg(\mathbb{X})$. Hence we obtain $\operatorname{ri}(\delta^{\sigma}_{\mathbb{X}})=k$ again. ∎

In the special case that $\mathbb{X}$ is a locally Gorenstein Cayley-Bacharach scheme, the regularity index of the Dedekind different attains the maximal value. This also follows from [14, Proposition 4.8] with a different proof.

Corollary 4.9**.**

In the setting of Proposition 4.8, assume that $\mathbb{X}$ is a Cayley-Bacharach scheme.

1. (a)

The regularity index of the Dedekind different $\delta^{\sigma}_{\mathbb{X}}$ is $2r_{\mathbb{X}}$. 2. (b)

The scheme $\mathbb{X}$ is arithmetically Gorenstein if and only if the Hilbert function of $\delta^{\sigma}_{\mathbb{X}}$ satisfies $\operatorname{HF}_{\delta^{\sigma}_{\mathbb{X}}}(i)=\operatorname{HF}_{\mathbb{X}}(i-r_{\mathbb{X}})$ for all $i\in\mathbb{Z}$.

Proof.

Claim (a) follows directly from the proposition, and claim (b) follows by [13, Proposition 5.8]. ∎

Acknowledgments.

The second author thanks the University of Passau for its hospitality and support during part of the preparation of this paper. The third and fourth authors would also like to acknowledge the support from the Vietnam National Foundation (TN-8).