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Search: All articles in the CJM digital archive with keyword Cohen-Macaulay

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1. CJM 2016 (vol 69 pp. 1274)

Favacchio, Giuseppe; Guardo, Elena
The Minimal Free Resolution of Fat Almost Complete Intersections in $\mathbb{P}^1\times \mathbb{P}^1$
A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of $I$. In this paper, we focus on the case that $I=I_X$ is an ideal defining an almost complete intersection (ACI) set of points $X$ in $\mathbb{P}^1 \times \mathbb{P}^1$. In particular, we describe a minimal free bigraded resolution of a non arithmetically Cohen-Macaulay (also non homogeneous) set $\mathcal Z$ of fat points whose support is an ACI, generalizing a result of S. Cooper et al. for homogeneous sets of triple points. We call $\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\geq 1.$

Keywords:points in $\mathbb{P}^1\times \mathbb{P}^1$, symbolic powers, resolution, arithmetically Cohen-Macaulay
Categories:13C40, 13F20, 13A15, 14C20, 14M05

2. CJM 2004 (vol 56 pp. 716)

Guardo, Elena; Van Tuyl, Adam
Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions
We study the Hilbert functions of fat points in $\popo$. If $Z \subseteq \popo$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for $l \gg 0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in $\popo$. This enables us to compute all but a finite number values of $H_{Z}$ without using the coordinates of points. We also characterize the ACM fat point schemes sing our description of the eventual behaviour. In fact, n the case that $Z \subseteq \popo$ is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Keywords:Hilbert function, points, fat points, Cohen-Macaulay, multi-projective space
Categories:13D40, 13D02, 13H10, 14A15

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