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Segre invariant and a stratification of the moduli space of coherent systems
International Journal of Mathematics ( IF 0.6 ) Pub Date : 2020-10-22 , DOI: 10.1142/s0129167x20501177
L. Roa-Leguizamón 1
Affiliation  

The aim of this paper is to generalize the [Formula: see text]-Segre invariant for vector bundles to coherent systems. Let [Formula: see text] be a non-singular irreducible complex projective curve of genus [Formula: see text] and [Formula: see text] be the moduli space of [Formula: see text]-stable coherent systems of type [Formula: see text] on [Formula: see text]. For any pair of integers [Formula: see text] with [Formula: see text], [Formula: see text] we define the [Formula: see text]-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the [Formula: see text]-Segre invariant induces a stratification of the moduli space [Formula: see text] into locally closed subvarieties [Formula: see text] according to the value [Formula: see text] of the function. We determine an above bound for the [Formula: see text]-Segre invariant and compute a bound for the dimension of the different strata [Formula: see text]. 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 [Formula: see text].

中文翻译:

Segre 不变量和相干系统模空间的分层

本文的目的是将向量束的 [公式:见文本]-Segre 不变量推广到相干系统。令[公式:见文]为属[公式:见文]的非奇异不可约复射影曲线,[公式:见文]为[公式:见文]的模空间-[公式:见文]类型的稳定相干系统: 见文本] 上 [公式:见文本]。对于任何一对整数 [Formula: see text] 和 [Formula: see text], [Formula: see text] 我们定义 [Formula: see text]-Segre 不变量,并证明它定义了族上的下半连续函数的连贯系统。因此,[公式:参见文本]-Segre 不变量根据函数的值 [公式:参见文本] 将模空间 [公式:参见文本] 分层为局部封闭的子变量 [公式:参见文本]。我们确定 [公式:见文本]-Segre 不变量的上述界限,并计算不同层 [公式:见文本] 的维度的界限。此外,我们给出了不同层非空的一些条件。为了证明上述结果,我们引入了子类型相干系统的概念[公式:见正文]。
更新日期:2020-10-22
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