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Resolutions of ideals of fat points with support in a hyperplane

Giuliana Fatabbi·Brian Harbourne·Anna Lorenzini
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Abstract

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 P^d. Assume that the given points lie in a hyperplane P^{d-1} in P^d, 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[P^d] in terms of the minimal graded free resolutions of the ideals I(j,d-1) in k[P^{d-1}] with j < m+1. As a corollary, we give the following formula for the Poincare polynomial P_{m,d} of I(m,d) in terms of the Poincare polynomials P_{j,d-1} of I(j,d-1): P_{m,d} = (1 + XT)(Σ_{0<j\le m} T^{m-j}(P_{j,d-1} - 1)) + 1 + XT^m.

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Giuliana Fatabbi, Brian Harbourne, Anna Lorenzini. 2005-07-11. Resolutions of ideals of fat points with support in a hyperplane. https://arxiv.org/abs/math/0502418

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