Classifying Hilbert functions of fat point subschemes in ℙ2

Abstract

The paper [10] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p 1+...+2p r, for all possible choices ofr distinct points in ℙ2. We study this problem forr points in ℙ2 over an algebraically closed fieldk of arbitrary characteristic in case eitherr ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [17, 18] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We sayp 1, ...,p r andp 1 ′, ... ,p r ′ have the sameconfiguration type if for all choices of nonnegative integersm i ,Z=m 1 p 1+...+m r p r andZ′=m 1 p 1 ′+...+m r p r ′ have the same Hilbert function.) Assuming either that 7 ≤r ≤ 8 (see [12] for the casesr ≤6) or that the pointsp i lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficientsm i determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) ofZ=m 1 p 1+...+m r p r . We demonstrate our results by explicitly listing all Hilbert functions for schemes ofr≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.