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Let $X$ be a smooth, projective curve of genus at least $4$ and $X$ non-hyperelliptic. I am looking for additional conditions on $X$ such that the Jacobian $J(X)$ of $X$ contains an unique principal polarization. Any idea/reference will be most welcome.

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    $\begingroup$ An obvious sufficient condition is that the Néron-Severi group $\operatorname{NS}(JX) $ is equal to $\Bbb{Z}$ (equivalently, the Picard number is equal to $1$). This is the case for a general curve of a given genus, and there are also explicit examples in every genus due to Mori (Japan. J. Math. (1977), no. 1, 105-109). I doubt that you can say more. By the way, the fact that $X$ is non-hyperelliptic or of genus $\geq 4$ is irrelevant. $\endgroup$
    – abx
    Aug 7, 2018 at 6:12

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