Hilbert series of generic ideals in products of projective spaces

TL;DR

This paper explores the Hilbert series of generic ideals in bigraded rings associated with products of projective spaces, extending conjectures from polynomial rings to more complex geometric settings.

Contribution

It introduces a conjecture on the Hilbert series of generic ideals in bigraded rings related to products of projective spaces, expanding existing theories beyond polynomial rings.

Findings

Proposes a new conjecture for Hilbert series in bigraded contexts.

Extends classical results from polynomial rings to products of projective spaces.

Provides a framework for future verification and research in algebraic geometry.

Abstract

If $I=(f_1,\ldots,f_r)$ is an ideal in $S=k[x_1,\ldots,x_n]$, and $f_i$ are "general" elements of given degrees, there is a conjecture on the Hilbert series of $S/I$. We are considering the corresponding concepts in bigraded rings.