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 $k=2$.

15 pages, accepted for publication in Acta Mathematica Vietnamica (AMV)