Moduli spaces of coherent systems of small slope on algebraic curves

TL;DR

This paper investigates the geometric properties of moduli spaces of coherent systems on algebraic curves, focusing on small slope cases, establishing their irreducibility and conditions for non-emptiness.

Contribution

It provides a detailed analysis of moduli spaces of coherent systems with small slope, proving irreducibility and characterizing non-empty cases.

Findings

Moduli spaces are irreducible when non-empty.

Necessary and sufficient conditions for non-emptiness are established.

Focus on cases where 0<d≤2n.

Abstract

Let $C$ be an algebraic curve of genus $g\ge2$. A coherent system on $C$ consists of a pair $(E,V)$, where $E$ is an algebraic vector bundle over $C$ of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of the space of sections of $E$. The stability of the coherent system depends on a parameter $\alpha$. We study the geometry of the moduli space of coherent systems for $0<d\le2n$. We show that these spaces are irreducible whenever they are non-empty and obtain necessary and sufficient conditions for non-emptiness.