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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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