Hodge polynomials of some moduli spaces of Coherent Systems

TL;DR

This paper investigates the structure of certain moduli spaces of coherent systems, providing stratifications, geometric descriptions, and explicit calculations of their Hodge polynomials, especially for specific low-rank cases.

Contribution

It introduces a stratification of the moduli space for coherent systems arising from BGN extensions with semistable quotients, and computes their Hodge polynomials in special cases.

Findings

Stratification of the moduli space into irreducible, smooth strata.

Descriptions of strata as complements of determinantal varieties.

Explicit computation of Hodge and Poincaré polynomials for specific cases.

Abstract

When $k<n$, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which $(n,d,k)=(3,d,1)$ and $d$ is even, obtaining from them the usual Poincar\'e polynomials.