There are many examples of pairs of non-isomorphic curves of genus 2 or 3 whose Jacobians are isomorphic as (unpolarized) Abelian varieties, see e.g. this post and its answers. This is relatively easy because almost all principally polarized Abelian varieties of dimension 2 and 3 are Jacobians. Ciliberto and Van der Geer have constructed families of examples in genus 4 (in Classication of Algebraic Varieties, Contemp. Math. 162, AMS (1994)). Does anyone know examples in genus $\geq 5$?
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