Segre invariant and a stratification of the moduli space of coherent systems

TL;DR

This paper extends the Segre invariant concept to coherent systems on algebraic curves, creating a stratification of the moduli space based on these invariants and analyzing the properties of these strata.

Contribution

The paper introduces the $(m,t)$-Segre invariant for coherent systems, generalizing previous invariants, and establishes a stratification of the moduli space with detailed properties.

Findings

The $(m,t)$-Segre invariant induces a semicontinuous stratification.

Conditions for non-empty strata are determined.

Dimensions of the strata are computed.

Abstract

The aim of this paper is to generalize the $m-$Segre invariant for vector bundles to coherent systems. Let $X$ be a non-singular irreducible complex projective curve of genus $g$ over $\mathbb{C}$ and $(E,V)$ be a coherent system on $X$ of type $(n,d,k)$. For any pair of integers $m, t$, $0 < m < n$, $0 \leq t \leq k$ we define the $(m,t)-$Segre invariant, denoted by $S^\alpha_{m,t}$ and show that $S^\alpha_{m,t}$ induces a semicontinuous function on the families of coherent systems. Thus, $S^\alpha_{m,t}$ gives a stratification of the moduli space $G(\alpha;n,d,k)$ of $\alpha-$stable coherent systems of type $(n,d,k)$ on $X$ into locally closed subvarieties $G(\alpha;n,d,k;m,t;s)$ according to the value $s$ of $S^\alpha_{m,t}$. We study the stratification, determine conditions under which the different strata are non-empty and compute their dimension.