Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes

Abstract

Given a fat point scheme W = m 1 P 1 + • • • + m s P s in the projective n-space P n over a field K of characteristic zero, the modules of Kähler differential k-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of W when k ∈ {1, . . . , n + 1}. In this paper we determine the value of its Hilbert polynomial explicitly for the case k = n + 1, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme Y = (m 1 −1)P 1 +• • •+(m s −1)P s . For n = 2, this allows us to determine the Hilbert polynomials of the modules of Kähler differential k-forms for k = 1, 2, 3, and to produce a sharp bound for the regularity index for k = 2.