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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.

##### MSC:
 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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