Determinant morphism for singular varieties

TL;DR

This paper extends the classical notion of the determinant of a coherent sheaf to singular projective varieties, analyzing its behavior in families and applications to moduli spaces and Hilbert schemes.

Contribution

It introduces a generalized determinant for coherent sheaves on singular varieties and studies its semi-continuous properties in families, connecting to moduli and Hilbert schemes.

Findings

Determinant defined for singular varieties matches classical in smooth case.

Hilbert polynomial of the determinant is upper semi-continuous in singular families.

Applications to moduli spaces of semi-stable sheaves and Hilbert schemes of curves.

Abstract

Let $X$ be a projective variety (possibly singular) over an algebraically closed field of any characteristic and $\mathcal{F}$ be a coherent sheaf. In this article, we define the determinant of $\mathcal{F}$ such that it agrees with the classical definition of determinant in the case when $X$ is non-singular. We study how the Hilbert polynomial of the determinant varies in families of singular varieties. Consider a singular family such that every fiber is a normal, projective variety. Unlike in the case when the family is smooth, the Hilbert polynomial of the determinant does not remain constant in singular families. However, we show that it exhibits an upper semi-continuous behaviour. Using this we give a determinant morphism defined over flat families of coherent sheaves. This morphism coincides with the classical determinant morphism in the smooth case. Finally, we give applications of our results to moduli spaces of semi-stable sheaves on $X$ and to Hilbert schemes of curves.