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

Abstract

Let X be a set of K-rational points in P 1 × 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 Kähler differentials Ω 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 × P 1 . Moreover, we introduce a Kähler different for Y and use it to characterize ACM reduced schemes in P 1 × P 1 having the Cayley-Bacharach property.