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Let $$X=\mathscr {J}(\widetilde{\mathscr {C}})$$ X = J ( C ~ ) , the Jacobian of a genus 2 curve $$\widetilde{\mathscr {C}}$$ C ~ over $${\mathbb {C}}$$ C , and let Y be the associated Kummer surface. Consider an ample line bundle $$L=\mathscr {O}(m\widetilde{\mathscr {C}})$$ L = O ( m C ~ ) on X for an even number m, and its descent to Y, say $$L'$$ L ′ . We show that any dominating component of $${\mathscr {W}}^1_{d}(|L'|)$$ W d 1 ( | L ′ | ) corresponds to $$\mu _{L'}$$ μ L ′ -stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve $$C\in |L|$$ C ∈ | L | and a base-point free $$g^1_d$$ g d 1 on C, say (A, V), we study the $$\mu _L$$ μ L -semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (A, V)) on X. Under certain assumptions on C and the $$g^1_d$$ g d 1 , we show that the above Lazarsfeld–Mukai bundles are $$\mu _L$$ μ L -semistable.
ANNALI DELL'UNIVERSITA' DI FERRARA – Springer Journals
Published: Mar 10, 2017
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