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

L. Roa-Leguizamón

Centro de Investigación en Matemáticas, A.C. (CIMAT), Apdo. Postal 402, C.P. 36240 Guanajuato, Gto, México

https://doi.org/10.1142/S0129167X20501177Cited by:0 (Source: Crossref)

Abstract

The aim of this paper is to generalize the $m$ $m$ -Segre invariant for vector bundles to coherent systems. Let $X$ $X$ be a non-singular irreducible complex projective curve of genus $g \geq 0$ $g \geq 0$ and $G (α; n, d, k)$ $G (α; n, d, k)$ be the moduli space of $α$ $α$ -stable coherent systems of type $(n, d, k)$ $(n, d, k)$ on $X$ $X$ . For any pair of integers $(m, t)$ $(m, t)$ with $0 < m < n$ $0 < m < n$ , $0 \leq t \leq k$ $0 \leq t \leq k$ we define the $(m, t)$ $(m, t)$ -Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the $(m, t)$ $(m, t)$ -Segre invariant induces a stratification of the moduli space $G (α; n, d, k)$ $G (α; n, d, k)$ into locally closed subvarieties $G (α; n, d, k; m, t; s)$ $G (α; n, d, k; m, t; s)$ according to the value $s$ $s$ of the function. We determine an above bound for the $(m, t)$ $(m, t)$ -Segre invariant and compute a bound for the dimension of the different strata $G (α; n, d, k; m, t; s)$ $G (α; n, d, k; m, t; s)$ . Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype $(a)$ $(a)$ .

Keywords:

AMSC: 14H60, 14D20

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