On the Semistability of certain Lazarsfeld-Mukai bundles on Abelian surfaces

Abstract

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

17 pages, revised version, statement of the main theorem has been revised, accepted for publication in Annali dell' Universita' di Ferrara

References in corpus (1)

• Movable curves and semistable sheaves