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 of r distinct points in P 2 . We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic in case either r ≤ 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 say p 1 , . . . , p r and p 1 , . . . , p r have the same configuration type if for all choices of nonnegative integers m i , Z = m 1 p 1 + · · · + m r p r and Z = m 1 p 1 + · · · + m r p r have 159 Collectanea Mathematica (electronic version): http://www.collectanea.ub.edu Geramita, Harbourne and Migliore the same Hilbert function.) Assuming either that 7 ≤ r ≤ 8 (see [12] for the cases r ≤ 6) or that the points p i lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients m i determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of Z = m 1 p 1 + · · · + m r p r . We demonstrate our results by explicitly listing all Hilbert functions for schemes of r ≤ 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.

Symbolic Powers Versus Regular Powers of Ideals of General Points in ℙ1× ℙ1

“…, which we will also denote by µ (i, j) . Using Lemma 2.3.2, the inequalities i ≥ j ≥ m ≥ 2, i + 2 j ≥ 5m, 2i + j ≥ 5m, and i + j > 3m show that A • B ≥ 0 for every exceptional curve B on X, and hence A is effective and nef (since for a blow up X of P 1 × P 1 at five multiplicity 1 generic points, and thus 6 general points of P 2 , using the results of [10] one checks that the only prime divisors of negative self-intersection are the exceptional curves, but any divisor meeting every exceptional curve non-negatively is effective and nef [10, Proposition 4.1]).…”

“…so for any prime divisor C we have D • C ≥ 0. By the adjunction formulaC 2 −C • D = 2p C − 2, we see that C 2 ≥ −2, with C • D = 1 if C 2 = −1 and C • D = 0 if C 2 = −2.There are only finitely many possible classes of reduced irreducible curves C with C • D = 0 when s ≤ 7 (see[10, Proposition 4.1]). For each of these classes, C is not effective if the points p i are general, so in fact no such C is effective if s ≤ 7 and the points p i are general.…”