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Search: MSC category 13A15 ( Ideals; multiplicative ideal theory )

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1. CJM Online first

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 2012 (vol 65 pp. 823)

Guardo, Elena; Harbourne, Brian; Van Tuyl, Adam
Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$
Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in $\mathbb{P}^3$ which can also be viewed as points in $\mathbb{P}^1 \times \mathbb{P}^1$. We also obtain results on the closely related problem, studied by Hochster and by Li-Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power.

Keywords:symbolic powers, multigraded, points
Categories:13F20, 13A15, 14C20

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