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Let $S$ be a surface defined over a field $K$, when is the albanese map $S\longrightarrow \text{Alb}(S)$ an embedding?

For curves, for example, with genus at least 2, there is a morphism between the symmetric square to the Jacobian, and it is an embedding whenever the curve is not hyperelliptic. In particular, this means that the morphism into the albanese is an embedding. I am curious if there is an intrinsic criterion.

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    $\begingroup$ $\Omega_S^1$ should be very ample in a suitable sense: it should be globally generated with enough sections to separate points. $\endgroup$
    – Donu Arapura
    Commented Sep 22, 2023 at 12:07
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    $\begingroup$ Just addding to Donu Arapura's comment: A nice reference in my opinion is Moriwaki's paper arxiv.org/abs/alg-geom/9410023 $\endgroup$
    – Ariyan Javanpeykar
    Commented Sep 22, 2023 at 13:51

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