Vector bundles with a fixed determinant on an irreducible nodal curve

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.Comment: 7 page

Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli space $U'$ of semistable vector bundles of rank $n$ and degree $d$ on an irreducible nodal curve $Y$ and show that $U'$ is locally factorial. We determine the canonical line bundles of $U'$ and $U'_L$, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification of $U'$.Comment: 16 pages, no figures, no table

Moduli spaces of vector bundles on a singular rational ruled surface

We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over $\mathbb R$.Comment: Final versio

Maps into projective spaces

We compute the cohomology of the Picard bundle on the desingularization J~d(Y) of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill–Noether loci in J~d(Y). We show that the moduli space M of morphisms of a fixed degree from Y to a projective space has a smooth compactification. As another application of the cohomology of the Picard bundle, we compute a top intersection number for the moduli space M confirming the Vafa–Intriligator formulae in the nodal case

Moduli of parabolic G-bundles

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Torsion free sheaves over a nodal curve of arithmetic genus one

We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined over C. Let X be a nodal curve of arithmetic genus one defined over R, with exactly one node, such that X does not have any real points apart from the node. We classify all isomorphism classes of stable real algebraic torsion free sheaves over X of even rank. We also classify all isomorphism classes of real algebraic torsion free sheaves over X of rank one

Generalized parabolic sheaves on an integral projective curve

We extend the notion of a parabolic vector bundle on a smooth curve to define generalized parabolic sheaves (GPS) on any integral projective curve X. We construct the moduli spaces M(X) of GPS of certain type on X. If X is obtained by blowing up finitely many nodes in Y then we show that there is a surjective birational morphism from M(X) to M(Y). In particular, we get partial desingularisations of the moduli of torsion-free sheaves on a nodal curve Y