Vector bundles with a fixed determinant on an irreducible nodal curve

Details Vector bundles with a fixed determinant on an irreducible nodal curve Vector bundles with a fixed determinant on an irreducible nodal curve

2005-12-14

arxiv.org/abs/math/0512318v1

Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4, November 2005, pp. 445-451

Mathematics

Algebraic Geometry

7 pages

Scientific paper

Let $M$ be the moduli space of generalized parabolic bundles (GPBs) of rank $r$ and degree $d$ on a smooth curve $X$. Let $M_{\bar L}$ be the closure of its subset consisting of GPBs with fixed determinant ${\bar L}$. We define a moduli functor for which $M_{\bar L}$ 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 $M_{\bar L}$ 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.

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