Curves of genus g lying on a g-dimensional Jacobian variety.

Fabio Bardelli; Gian Pietro Pirola

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We give a parametrization of curves C of genus 2 with a maximal isotropic (ℤ/3)² in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial 3-part of the Tate-Shafarevich group.

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In the moduli space of curves of genus $g$ , $ℳ_{g}$ , let ${𝒢𝒫}_{g}$ be the locus of curves that do not satisfy the Gieseker-Petri theorem. In the genus seven case we show that ${𝒢𝒫}_{7}$ is a divisor in $ℳ_{7}$ .