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Fat points in \(\mathbb{P}^1\times\mathbb{P}^1\) and their Hilbert functions. (English) Zbl 1092.14057
The coordinate ring of \(\mathbb P^1\times\mathbb P^1\) is the bigraded ring \(R = k[x_0,x_1,y_0,y_1]\) where \(\deg(x_i) = (1,0)\), \(\deg(y_j)=(0,1)\) and \(k\) is an algebraically closed field. A fat point scheme in \(\mathbb P^1\times\mathbb P^1\) is a subscheme \(Z\) defined by an ideal \(I_Z ={\mathfrak p}^{m_1}_1\cap \cdots \cap {\mathfrak p}^{m_s}_s\) where \({\mathfrak p}_i\) is the ideal defining a reduced point and \(m_i\geq 1\). The authors study the Hilbert function \(H_Z(i,j)=\dim_kR_{i,j}/(I_Z)_{i,j}\) of such schemes \(Z\).
Since the coordinate ring of \(Z\) contains homogeneous non-zero divisors of degree \((1,0)\) and \((0,1)\), they can easily show that \(H_Z\) is non-decreasing and eventually constant in each row and colum. A much less obvious result is that the eventual values of \(H_Z\) in each row and column can be calculated directly from the multiplicities of the points and the relative positions of the points in the support of \(Z\) with respect to the lines of the two rulings.
Next, the authors use the eventual behaviour of \(H_Z\) to derive further information about the scheme \(Z\). In particular, they prove that the arithmetically Cohen-Macaulay (ACM) property (i.e. the property that \(R/I_Z\) is a Cohen-Macaulay ring) can be checked using these eventual values. They also derive a number of further informations about the multiplicities and relative position of the points. The paper concludes with some special configurations of ACM fat point schemes.

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