The family G_T of graded quotients of k[x,y] of given Hilbert function

Abstract

The nonsingular variety G_T parametrizes all graded ideals I of R=k[x,y] for which the Hilbert function H(R/I)=T. The variety G_T has a natural cellular decomposition: each cell V(E) corresponds to a monomial ideal E for which H(R/E)=T. Given a monomial ideal E, the quotient R/E has a basis of monomials in the shape of a partition P(E), having "diagonal lengths" T. The dimension of the cell V(E) is the number of difference one hooks of P(E). We show that G_T is birational to a certain product SGrass(T) of small Grassman varieties, and that over the complexes, the birational map induces an isomorphism of homology groups, but not usually an isomorphism of rings. We determine the homology ring) when T=(1,2,...,d-1,d,...,d,1). This G(T) is the desingularisation of the d-secant bundle Sec(d,j) of the degree-j rational normal curve, We determine the classes of the pullbacks of the higher singular loci of Sec(d,j). We also use the ring H*(G_T) to determine the number of ideals satisfying an intersection of ramification conditions at different points of P^1. A main tool is a combinatorial "hook code" for the partition P(E), that gives the image of the cell V(E) in SGrass(T).