2005/02/19 by Giuliana Fatabbi, Fatabbi, Giuliana, Brian Harbourne +3
Mathematics · #13D02 #14Q99 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13D02 #msc:14Q99
paper · pdf · doi:10.48550/arxiv.math/0502418
10 pages; to appear in Proc. Amer. Math. Soc.; some expositional changes; added a reference to paper of Geramita, Migliore and Sabourin (math.AC/0411445)
arxiv created 2005/07/11 · arxiv updated 2009/12/01
Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplane. Suppose, for example, that I(m,d) is the ideal defining r given points of multiplicity m in the projective space Pd. Assume that the given points lie in a hyperplane Pd-1 in Pd, and that the ground field k is algebraically closed of characteristic 0. We give an explicit minimal graded free resolution of I(m,d) in k[Pd] in terms of the minimal graded free resolutions of the ideals I(j,d-1) in k[Pd-1] with j < m+1. As a corollary, we give the following formula for the Poincare polynomial Pm,d of I(m,d) in terms of the Poincare polynomials Pj,d-1 of I(j,d-1): Pm,d = (1 + XT)(Σ0<j≤ m Tm-j(Pj,d-1 - 1)) + 1 + XTm.