Usha N. Bhosle

Parabolic vector bundles on curves

Seshadri introduced the notion of parabolic structures on vector bundles [4] and later constructed a moduli space for semistable parabolic vector bundles on curves [2]. In this small note we describe a different construction of the moduli space generalising the method of Gieseker [1]. This has some advantages. This construction is much simpler and shorter than in [2]. It avoids the use of unitary bundles and hence is applicable in positive characteristics. One does not need the introduction and comparison of different parabolic structures. Moreover, some computations which have to be repeated here (proposition 2) become simpler in this method. The generalisation to parabolic principal bundles will be considered in a subsequent paper. I would like to thank M. S. Narasimhan and A. Ramanathan for helpful discussions.

Generalized parabolic bundles and applications— II

We prove the existence of the moduli spaceM(n,d) of semistable generalised parabolic bundles (GPBs) of rankn, degreed of certain general type on a smooth curve. We study interesting cases of the moduli spacesM(n, d) and find explicit geometric descriptions for them in low ranks and genera. We define tensor products, symmetric powers etc. and the determinant of a GPB. We also define fixed determinant subvarietiesML(n, d),L being a GPB of rank 1. We apply these results to study of moduli spaces of torsionfree sheaves on a reduced irreducible curveY with nodes and ordinary cusps as singularities. We also study relations among these moduli spaces (rank 2) as polarization varies over [0, 1].

On linear series and a conjecture of D. C. Butler

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H-0 (L) of dimension n+1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V circle times O-C -> L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

Stable real algebraic vector bundles over a Klein bottle

Let X be a geometrically connected smooth projective curve of genus one, defined over the field of real numbers, such that X does not have any real points. We classify the isomorphism classes of all stable real algebraic vector bundles over X.

Brill-Noether theory on nodal curves

We study the Brill–Noether loci for semistable torsionfree sheaves of small slopes on a nodal curve. We determine conditions for nonemptiness and irreducibility of these loci.

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 spacesM(X) of GPS of certain type onX. IfX is obtained by blowing up finitely many nodes inY 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 curveY.

Vector bundles with a fixed determinant on an irreducible nodal curve

LetM be the moduli space of generalized parabolic bundles (GPBs) of rankr and degree dona smooth curveX. LetM−L be the closure of its subset consisting of GPBs with fixed determinant− L. We define a moduli functor for whichM−L is the coarse moduli scheme. Using the correspondence between GPBs onX and torsion-free sheaves on a nodal curveY of whichX is a desingularization, we show thatM−L can be regarded as the compactified moduli scheme of vector bundles onY 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 onY. The relation to Seshadri-Nagaraj conjecture is studied.

Tensor fields and singular principal bundles

Let G be a complex reductive algebraic group. We construct the moduli spaces of tensor fields of special type on a large class of varieties X including seminormal varieties and varieties satisfying Serres condition S 2 . We use them to construct compactified moduli spaces of principal G-bundles on X in case X is irreducible. We also give a simpler construction of the moduli spaces of orthogonal and symplectic sheaves on curves with ordinary nodes and cusps.

Brauer Group and Birational Type of Moduli Spaces of Torsionfree Sheaves on a Nodal Curve

Let be the moduli space of stable vector bundles of rank n and fixed determinant 𝕃 of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree d will be denoted by . We calculate the Brauer groups of . We study the question of rationality of and .

Singular pencils of quadrics and compactified Jacobians of curves

AbstractLetY be an irreducible nodal hyperelliptic curve of arithmetic genusg such that its nodes are also ramification points (char ≠2). To the curveY, we associate a family of quadratic forms which is dual to a singular pencil of quadrics in