Vector bundles with a fixed determinant on an irreducible nodal curve

Abstract

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

Generators of the cohomology ring, after Newstead

“…Note that if E is locally free then this condition implies ∧ 2 E ∼ = L 0 . By [31,Theorem 1], there exists a moduli space, denoted U X 0 (2, L 0 ), parameterizing rank 2, semi-stable sheaves E on X 0 such that E has determinant L 0 (see also [4]). By [31,Theorem 3.7(3)], there exists a natural proper morphism θ :…”

On finite generation of the section ring of the determinant of cohomology line bundle

“…The Nagaraj-Seshadri locus consists of those bundles which arise as the fiber of the closure over t = 0 for such families. A conjectural description of the underlying reduced set of this locus, given in [NS97, Conjecture page 136], was shown to hold by Sun in [Sun02,Sun03] (also see [NR93,Bho05,Bho99]). Sun [Sun03] also considered the case C 0 is reducible, and showed that in this case the Nagaraj-Seshadri locus is possibly reducible.…”

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

“…If E is locally free then this means ∧ 2 E ∼ = L 0 . Using [31,Theorem 2], one can check that there exists a moduli space, denoted U X 0 (2, L 0 ), parametrizing rank 2, semi-stable sheaves on X 0 with determinant L 0 (see also [9]). However, the moduli space U X 0 (2, L 0 ) is singular.…”