Vector Bundles with a Fixed Determinant on an Irreducible Nodal Curve

[摘要] Let 𝑀 be the moduli space of generalized parabolic bundles (GPBs) of rank 𝑟 and degree 𝑑 on a smooth curve 𝑋. Let $M_{overline{L}}$ be the closure of its subset consisting of GPBs with fixed determinant $overline{L}$. We define a moduli functor for which $M_{overline{L}}$ is the coarse moduli scheme. Using the correspondence between GPBs on 𝑋 and torsion-free sheaves on a nodal curve 𝑌 of which 𝑋 is a desingularization, we show that $M_{overline{L}}$ can be regarded as the compactified moduli scheme of vector bundles on 𝑌 with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves on 𝑌. The relation to Seshadri–Nagaraj conjecture is studied.

[学科分类] 数学（综合）

[关键词] Nodal curves;torsion-free sheaves;fixed determinant.