Articles written in Proceedings – Mathematical Sciences

    • Singular pencils of quadrics and compactified Jacobians of curves

      Usha N Bhosle

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      LetY 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$$\mathbb{P}^{2g + 1} $$ with Segre symbol [2...21...1], where the number of 2's is equal to the number of nodes. We show that the compactified Jacobian ofY is isomorphic to the spaceR of (g−1) dimensional linear subspaces of$$\mathbb{P}^{2g + 1} $$ which are contained in the intersectionQ of quadrics of the pencil. We also prove that (under this isomorphism) the generalized Jacobian ofY is isomorphic to the open subset ofR consisting of the (g−1) dimensional subspaces not passing through any singular point ofQ.

    • PrincipalG-bundles on nodal curves

      Usha N Bhosle

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      LetG be a connected semisimple affine algebraic group defined over C. We study the relation between stable, semistable G-bundles on a nodal curveY and representations of the fundamental group ofY. This study is done by extending the notion of (generalized) parabolic vector bundles to principal G-bundles on the desingularizationC ofY and using the correspondence between them and principal G-bundles onY. We give an isomorphism of the stack of generalized parabolic bundles onC with a quotient stack associated to loop groups. We show that if G is simple and simply connected then the Picard group of the stack of principal G-bundles onY is isomorphic to ⊕m Z,m being the number of components ofY.

    • Weak point property and sections of Picard bundles on a compactified Jacobian over a nodal curve


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      We show that the compactified Jacobian (and its desingularization) of an integral nodal curve $Y$ satisfies the weak point property and the Jacobian of $Y$ satisfies the diagonal property. We compute some cohomologies of Picard bundles on the compactified Jacobian and its desingularization

    • Picard bundle on the moduli space of torsionfree sheaves


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      Let $Y$ be an integral nodal projective curve of arithmetic genus $g\geq2$ with $m$ nodes defined over an algebraically closed field. Let $n$ and $d$ be mutually coprime integers with $n\geq2$ and $d$ > $n(2g −2)$. Fix a line bundle $L$ of degree $d$ on $Y$ .We prove that the Picard bundle $E_{L}$ over the ‘fixed determinant moduli space’ $U_{L}(n, d)$ is stable with respect to the polarisation $\theta_{L}$ and its restriction to the moduli space $U'_{L}(n, d)$, of vector bundles of rank $n$ and determinant $L$, is stable with respect to any polarisation. There is an embedding of the compactified Jacobian $\bar{J}(Y)$ in the moduli space $U_{Y}(n, d)$ of rank $n$ and degree $d$. We show that the restriction of the Picard bundle of rank $ng$ (over $U_{Y}(n, n(2g − 1)))$ to $\bar{J}(Y)$ is stable with respect to any theta divisor $\theta_{\bar{J}(Y)}$.

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