Kahler differentials for fat points schemes in P1xP1

Abstract. Let X be a set of K-rational points in P1 x P1 over a feld K ofcharacteristic zero, let Y be a fat point scheme supported at X, and let RYbe the bihomogeneus coordinate ring of Y. In this paper we investigate themodule of Kahler differentials Omega^1_RY=K. We describe this bigraded RY-moduleexplicitly via a homogeneous short exact sequence and compute its Hilbertfunction in a number of special cases, in particular when the support X is acomplete intersection or an almost complete intersection in P1xP1. Moreover,we introduce a Kahler different for Y and use it to characterize reduced fatpoint schemes in P1 x P1 having the Cayley-Bacharach property.