ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 64, Issue 1, pp 145–164 | Cite as

On the semistability of certain Lazarsfeld–Mukai bundles on abelian surfaces

Article

Abstract

Let \(X=\mathscr {J}(\widetilde{\mathscr {C}})\), the Jacobian of a genus 2 curve \(\widetilde{\mathscr {C}}\) over \({\mathbb {C}}\), and let Y be the associated Kummer surface. Consider an ample line bundle \(L=\mathscr {O}(m\widetilde{\mathscr {C}})\) on X for an even number m, and its descent to Y, say \(L'\). We show that any dominating component of \({\mathscr {W}}^1_{d}(|L'|)\) corresponds to \(\mu _{L'}\)-stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve \(C\in |L|\) and a base-point free \(g^1_d\) on C, say (AV), we study the \(\mu _L\)-semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (AV)) on X. Under certain assumptions on C and the \(g^1_d\), we show that the above Lazarsfeld–Mukai bundles are \(\mu _L\)-semistable.

Keywords

Semistability Hyperelliptic Jacobians 

Mathematics Subject Classification

14C20 14C21 14J60 14.51 

Notes

Acknowledgements

I thank Dr. Jaya NN Iyer, IMSc Chennai for introducing me to this problem and for her guidance at every stage of this project. I also thank Dr. T. E. Venkata Balaji, IIT Madras and Prof. D. S. Nagaraj, IMSc Chennai for helpful discussions and their support. I also thank Prof. Robert Lazarsfeld and Prof. Jason Starr at Stony Brook University for useful comments on an earlier version of the manuscript.

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Copyright information

© Università degli Studi di Ferrara 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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