Free Resolutions of Fat Point Ideals on P

Given distinct points p1, . . . , pr of a smooth variety V (over an algebraically closed field k) and positive integers mi, Z = m1p1 + · · · +mrpr denotes the subscheme defined locally at each point pi by I mi i , where Ii is the maximal ideal in the local ring OV,pi at pi of the structure sheaf. More briefly, we say Z is a fat point subscheme of V . In the case that V is P for some n, it is of interest to study the homogeneous ideal IZ defining Z as a subscheme of P ; IZ is called an ideal of fat points. Given an ideal IZ of a fat point subscheme Z ⊂ P, one first may want to determine its Hilbert function hIZ , defined for each d by hIZ (d) = dimk((IZ )d), where (IZ)d is the homogeneous component of IZ of degree d (i.e., (IZ)d is the k-vector space of all homogeneous forms of degree d in IZ). One next may wish to study a minimal free resolution · · · → F1 → F0 → IZ → 0 of IZ , beginning with determining F0. Determining F0 as a graded module over the homogeneous coordinate ring k[P] of P is equivalent to finding the number νd(Z) of generators in each degree d in a minimal homogeneous set of generators for IZ , since F0 = ⊕