• Usha N Bhosle

      Articles written in Proceedings – Mathematical Sciences

    • Generalized parabolic sheaves on an integral projective curve

      Usha N Bhosle

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      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.

    • Generalized parabolic bundles and applications— II

      Usha N Bhosle

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      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].

    • Picard groups of the moduli spaces of semistable sheaves I

      Usha N Bhosle

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      We compute the Picard group of the moduli spaceU′ of semistable vector bundles of rankn and degreed on an irreducible nodal curveY and show thatU′ is locally factorial. We determine the canonical line bundles ofU′ andUL, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification ofU′.

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

      Usha N Bhosle

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      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.

    • Torsionfree Sheaves over a Nodal Curve of Arithmetic Genus One

      Usha N Bhosle Indranil Biswas

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      We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined over $\mathbb{C}$. Let 𝑋 be a nodal curve of arithmetic genus one defined over $\mathbb{R}$, 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 torsionfree sheaves over 𝑋 of even rank. We also classify all isomorphism classes of real algebraic torsionfree sheaves over 𝑋 of rank one.

    • Maps into Projective Spaces

      Usha N Bhosle

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      We compute the cohomology of the Picard bundle on the desingularization $\overline{J}^d (Y)$ of the compactified Jacobian of an irreducible nodal curve 𝑌. We use it to compute the cohomology classes of the Brill–Noether loci in $\overline{J}^d(Y)$.

      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.

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