Kaehler differentials for fat point schemes in P^1xP^1

TL;DR

This paper studies the module of Kähler differentials for fat point schemes in P^1 x P^1, providing explicit descriptions, Hilbert function computations, and characterizations of schemes with the Cayley-Bacharach property.

Contribution

It offers an explicit description of the Kähler differentials module for fat point schemes in P^1 x P^1 and characterizes schemes with the Cayley-Bacharach property.

Findings

Explicit description of the Kähler differentials module via a short exact sequence.

Computed Hilbert functions for special cases like complete intersections.

Characterization of schemes with the Cayley-Bacharach property.

Abstract

Let $X$ be a set of $K$-rational points in $P^1 \times P^1$ over a field $K$ of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, and let $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper we investigate the module of Kaehler differentials $\Omega^1_{R_Y/K}$. We describe this bigraded $R_Y$-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $X$ is a complete intersection or an almost complete intersection in $P^1 \times P^1$. Moreover, we introduce a Kaehler different for $Y$ and use it to characterize reduced fat point schemes in $P^1 \times P^1$ having the Cayley-Bacharach property.