The minimal free resolution of fat almost complete intersections in $\mathbb{P}^1\times\mathbb{P}^1$

TL;DR

This paper provides a detailed minimal free resolution for fat almost complete intersection point sets in 11, revealing that their symbolic and ordinary powers coincide, and extends understanding of their algebraic structure.

Contribution

It introduces the explicit minimal free bigraded resolution for non Cohen-Macaulay fat ACI point sets in 11 and proves the equality of symbolic and ordinary powers.

Findings

Explicit minimal free resolution for fat ACI points in 11.

Proves symbolic and ordinary powers are equal for these sets.

Enhances understanding of algebraic properties of fat point schemes.

Abstract

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) sets 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 homogeneus) set of fat points $\mathcal Z$ whose support is an ACI. 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 $m\geq 1.$