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Separators of arithmetically Cohen-Macaulay fat points $$\mathbb P^1\times \mathbb P^1$$. (English) Zbl 1285.13019
Fix an algebrically closed field $$k$$ of characteristic zero. Let $$R=k[x_0,x_1,y_0,y_1]$$ be the $$\mathbb{N}^2-$$graded polynomial ring with $$\deg(x_i)=(1,0)$$ for $$i=0,1$$ and $$\deg(y_i)=(0,1)$$ for $$i=0,1$$. The ring $$R$$ is the coordinate ring of $$\mathbb{P}^1 \times \mathbb{P}^1$$. Consider now a set of points $$X=\{P_1, \dots, P_s\} \subset \mathbb{P}^1 \times \mathbb{P}^1$$ and fix positive integers $$m_1, \dots, m_s$$. In this paper the authors study some of the properties of the scheme $$Z=m_1P_1+\cdots m_sP_s$$ of fat points. In particular, the main theorem of the paper shows how to compute the degree of a separator of $$P_i$$ of multiplicity $$m_i$$ directly from the combinatorics of the scheme $$Z$$, provided $$Z$$ is ACM.
Before to state such theorem we need to recall some definition.
Let $$X$$ be a set of $$s$$ distinct points in $$\mathbb{P}^1 \times \mathbb{P}^1$$. Let $$\pi_1:\mathbb{P}^1 \times \mathbb{P}^1 \rightarrow \mathbb{P}^1$$ denote the projection morphism defined by $$P \times Q \mapsto P$$. Let $$\pi_2:\mathbb{P}^1 \times \mathbb{P}^1 \rightarrow \mathbb{P}^1$$ be the other projection morphism. The set $$\pi_1(X) = \{P_1,\ldots,P_a\}$$ is the set of $$a \leq s$$ distinct first coordinates that appear in $$X$$. Similarly, $$\pi_2(X) = \{Q_1,\ldots,Q_b\}$$ is the set of $$b \leq s$$ distinct second coordinates. For $$i = 1,\ldots,a$$, let $$L_{P_i}$$ be the degree $$(1,0)$$ form that vanishes at all the points with first coordinate $$P_i$$. Similarly, for $$j = 1,\ldots,b$$, let $$L_{Q_j}$$ denote the degree $$(0,1)$$ form that vanishes at points with second coordinate $$Q_j$$.
Let $$D:=\{(x,y) ~|~ 1 \leq x \leq a, 1 \leq y \leq b\}.$$ If $$P \in X$$, then $$I_{P} = (L_{R_i},L_{Q_j})$$ for some $$(i,j) \in D.$$ So, we can write each point $$P \in X$$ as $$P_i \times Q_j$$ for some $$(i,j)\in D$$.
Definition. Suppose that $$X$$ is a set of distinct points in $$\mathbb{P}^1 \times \mathbb{P}^1$$ and $$|\pi_1(X)| = a$$ and $$|\pi_2(X)| = b$$. Let $$I_{P_i \times Q_j} = (L_{R_i},L_{Q_j})$$ denote the ideal associated to the point $$P_i \times Q_j \in X$$. For each $$(i,j) \in D$$, let $$m_{ij}$$ be a positive integer if $$P_i \times Q_j \in X$$, otherwise, let $$m_{ij} = 0$$. Then we denote by $$Z$$ the subscheme of $$\mathbb{P}^1 \times \mathbb{P}^1$$ defined by the saturated bihomogeneous ideal $Iz = \bigcap_{(i,j) \in D} I_{P_i\times Q_j}^{m_{ij}}$ We say $$Z$$ is a fat point scheme or a set of fat points of $$\mathbb{P}^1 \times \mathbb{P}^1$$. The integer $$m_{ij}$$ is called the multiplicity of the point $$P_i \times Q_j$$. The support of $$Z$$, written $$\mathrm{supp}(Z)$$, is the set of points $$X$$.

Definition. A fat point scheme is said to be arithmetically Cohen-Macaulay (ACM for short) if the associated coordinate ring is Cohen-Macaulay.

Definition. Let $$Z = m_1P_1 + \cdots + m_iP_i + \cdots + m_sP_s$$ be a set of fat points in $$\mathbb{P}^1 \times \mathbb{P}^1$$. We say that $$F$$ is a separator of the point $$P_i$$ of multiplicity $$m_i$$ if $$F \in I_{P_i}^{m_i-1} \setminus I_{P_i}^{m_i}$$ and $$F \in I_{P_j}^{m_j}$$ for all $$j \neq i$$.
If we let $$Z' = m_1P_1 + \cdots + (m_i-1)P_i + \cdots + m_sP_s$$, then a separator of the point $$P_i$$ of multiplicity $$m_i$$ is also an element of $$F \in I_{Z'}\setminus I_Z$$. The set of minimal separators are defined in terms of the ideals $$I_{Z'}$$ and $$I_Z$$.

Definition. A set $$\{F_1,\ldots,F_p\}$$ is a set of minimal separators of $$P_i$$ of multiplicity $$m_i$$ if $$I_{Z'}/I_Z = (\overline{F}_1,\ldots,\overline{F}_p)$$, and there does not exist a set $$\{G_1,\ldots,G_q\}$$ with $$q < p$$ such that $$I_{Z'}/I_Z = (\overline{G}_1,\ldots,\overline{G}_q)$$.

Definition. The degree of the minimal separators of $$P_i$$ of multiplicity $$m_i$$, denoted $$\deg_Z(P_i)$$, is the tuple $\deg_Z(P_i) = (\deg F_1,\ldots,\deg F_p)$
where $$\deg F_i \in \mathbb N^2$$ and $$F_1,\ldots,F_p$$ is any set of minimal separators of $$P_i$$ of multiplicity $$m_i$$.
For a general fat point scheme $$Z\subset \mathbb{P}^1 \times \mathbb{P}^1$$, there is no known formula for $$p=|\deg(P)|$$. However, if $$Z$$ is ACM, then $$p$$ can be computed. As a matter of fact, the main result of this paper is a formula to compute the degree of a minimal separator for each fat point in an ACM fat point scheme in $$\mathbb{P}^1 \times \mathbb{P}^1$$.
Theorem Let $$Z \subset \mathbb{P}^1 \times \mathbb{P}^1$$ be an ACM set of fat points with $$a=|\pi_1(\mathrm{supp}(Z))|$$ and $$b=|\pi_2(\mathrm{supp}(Z))|$$. Suppose $$P_i \times Q_j \in \mathrm{supp}(Z)$$ is a point with multiplicity $$m_{ij}$$. Set $a_{\ell} = \sum_{s=1}^a (m_{sj} - \ell)_+$ and $b_{\ell} = \sum_{p=1}^b (m_{ip} - \ell)_+$ for $$\ell = 0,\ldots,m_{ij}-1$$, where $$(m-t)_+=\max\{0, m-t\}$$. Then
$\deg_Z(P_i \times Q_j) = \{(a_{m_{ij}-1-\ell}-1,b_{\ell}-1) ~|~ \ell = 0,\ldots,m_{ij}-1\}.$

MSC:
 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D02 Syzygies, resolutions, complexes and commutative rings 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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