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Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes
Martin Kreuzer
,
Tran N. K. Linh
,
Le Ngoc Long
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 $k=2$.