Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes

Abstract

Given a fat point scheme \(\mathbb {W}=m_{1}P_{1}+\cdots +m_{s}P_{s}\) in the projective n-space \(\mathbb {P}^{n}\) 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 \(\mathbb {W}\) when \(k\in \{1,\dots , n+1\}\). 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 \(\mathbb {Y} = (m_{1}-1)P_{1} + {\cdots } + (m_{s}-1)P_{s}\). 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.