Kähler differentials and Kähler differents for fat point schemes

Given a fat point scheme W=m1P1+⋯+msPsW=m1P1+⋯+msPs in a projective space PnPn over a field K  , we study the module of Kähler differentials and the Kähler differents of its homogeneous coordinate ring RWRW. We describe the Hilbert functions and Hilbert polynomials of these objects and bound their index of regularity. For special cases, in particular if the support of WW is a complete intersection or has some kind of uniformity, or if n=4n=4, we present more detailed results, including proofs of the Segre bound for certain fat point schemes in P4P4.