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https://www.ias.ac.in/article/fulltext/pmsc/130/0034

Nodal curve; moduli spaces; Picard bundles; stability.

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 $\theta_{\bar{J}(Y)}$.

USHA N BHOSLE¹

1. Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

Published

11 June 2020

Manuscript received

14 March 2019

Manuscript revised

21 January 2020

Accepted

29 January 2020

Early published

Unedited version published online

Final version published online

6 June 2020