Betti strata of height two ideals

Abstract

Let R = k [ x , y ] denote the polynomial ring in two variables over an infinite field k. We study the Betti strata of the family G ( H ) parametrizing graded Artinian quotients of R = k [ x , y ] having given Hilbert function H. The Betti stratum G β ( H ) parametrizes all quotients A of having the graded Betti numbers determined by H and the minimal relation degrees β, with β i = dim k Tor 1 R ( I , k ) i . We recover that the Betti strata are irreducible, and we calculate their codimension in the family G ( H ) . Theorem The codimension of G β ( H ) in G ( H ) satisfies, letting ν i = # { generators of degree i } , and β i = # { relations of degree i } , cod G β ( H ) = ∑ i > μ ( β i ) ( ν i ) . Here μ = min { i | H i < i + 1 } , the initial degree of the ideals. When k is algebraically closed, we also show that the closure of a Betti stratum is the union of more special strata, and is Cohen–Macaulay (Theorem 2.18). Our method is to identify G β ( H ) as a product of determinantal varieties. Ours is a more direct argument than that of [A. Iarrobino, V. Kanev, Power Sums, Gorenstein Algebras, and Determinantal Loci, Lecture Notes in Math., vol. 1721, Springer, Heidelberg, 1999, 345+xxvii pp., Theorem 5.63], which obtains the codimension of Betti substrata of a postulation stratum for punctual subschemes of P 2 ; there we relied on a result of M. Boij determining the codimension of Betti strata of height three Gorenstein algebras [M. Boij, Betti number strata of the space of codimension three Gorenstein Artin algebras, preprint, 2001]. Key tools throughout include properties of an invariant τ ( V ) , the number of generators of the ancestor ideal V ¯ ⊂ R , and previous results concerning the projective variety G ( H ) in [A. Iarrobino, Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10 (188) (1977)]. We adapt a method of M. Boij that reduces the calculation of the codimension of the Betti strata to showing that the most special stratum has the right codimension. As applications we determine which Hilbert functions are possible for Artinian quotients of R, given the socle type (Theorem 3.2), and we also determine the Hilbert function of the intersection of t general enough level ideals in R, each having a specified Hilbert function (Theorem 3.6).