Cohomology bounds for sheaves of dimension one

TL;DR

This paper establishes optimal bounds on the number of global sections for one-dimensional semistable sheaves on projective varieties, generalizing Clifford's theorem and exploring the structure of moduli spaces.

Contribution

It introduces a method to determine sharp bounds on $h^0(F)$ using the spectrum of sheaves, extending classical results to higher-dimensional varieties.

Findings

Derived sharp bounds for $h^0(F)$ on projective varieties.

Connected the deepest stratum of the moduli space to a subscheme of a relative Hilbert scheme.

Provided examples of semistable sheaves with maximal global sections.

Abstract

We find the sharp bounds on $h^0(F)$ for one-dimensional semistable sheaves $F$ on a projective variety $X$ by using the spectrum of semistable sheaves. The result generalizes the Clifford theorem. When $X$ is the projective plane $\mathbb{P}^2$, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a subscheme of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.