2020/03/24 by Martin Kreuzer, Kreuzer, Martin, Tran N. K. Linh +3
Mathematics · #13D40 #13N05 #14C99 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13D40 #msc:13N05 #msc:14C99
paper · pdf · doi:10.48550/arxiv.2003.11390
15 pages, accepted for publication in Acta Mathematica Vietnamica (AMV)
arxiv created 2020/08/12 · arxiv updated 2020/08/13
Given a fat point scheme \mathbbW=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 \mathbbW 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 \mathbbY = (m1-1)P1 + ⋯ + (ms-1)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.