Hilbert series of generic ideals in products of projective spaces

Abstract

If k[x 1 , . . . , x n ]/I = R = i≥0 R i , k a field, is a standard graded algebra, the Hilbert series of R is the formal power series i≥0 dim k R i t i . It is known already since Macaulay which power series are Hilbert series of graded algebras [12]. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals I = (f 1 , . . . , f r ), deg f i = d i , i = 1, . . . , r. In some sense "most" ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those "generic" ideals, see below. In this paper we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of P 1 × P 1 , which might be easier to prove.