Two-coverings of Jacobians of Curves of Genus Two
نویسندگان
چکیده
Given a curve C of genus 2 defined over a field k of characteristic different from 2, with Jacobian variety J , we show that the two-coverings corresponding to elements of a large subgroup of H ` Gal(k/k), J [2](k) ́ (containing the Selmer group when k is a global field) can be embedded as intersection of 72 quadrics in P k , just as the Jacobian J itself. Moreover, we actually give explicit equations for the models of these twists in the generic case, extending the work of Gordon and Grant which applied only to the case when all Weierstrass points are rational. In addition, we describe elegant equations on the Jacobian itself, and answer a question of Cassels and the first author concerning a map from the Kummer surface in P to the desingularized Kummer surface in P.
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