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Acta Mathematica Vietnamica

Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes

Martin Kreuzer , icon-email Tran N. K. Linh , Le Ngoc Long

Abstract

Given a fat point scheme W=m1P1++msPs in the projective n-space Pn 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=(m11)P1++(ms1)Ps. 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.