Vector bundles with a fixed determinant on an irreducible nodal curve

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