Kähler Differentials for Fat Point Schemes in ℙ1×ℙ1

Abstract

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