### On a Theorem of Castelnuovo, and the Equations Defining Space Curves.

L. Gruson, C. Peskine, R. Lazarsfeld (1983)

Inventiones mathematicae

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L. Gruson, C. Peskine, R. Lazarsfeld (1983)

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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.