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In J.P. Murre's "Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of mumford", in the proof of Theorem 3.11 he remarks that "the Jacobian of a curve is ’irreducible’ as a principally polarized abelian variety (i.e. does not split up in a product of principally polarized abelian varieties)."

I was not able to find any reference to an explanation or proof of this result. Can anyone point one out to me, or explain why this result is true? Much appreciated!

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    $\begingroup$ If an abelian variety is reducible, then the theta divisor is reducible as a variety. But the theta divisor of a Jacobian is irreducible, because is the image of a symmetric product of a curve. $\endgroup$
    – Daniele A
    Commented Dec 13, 2021 at 11:51
  • $\begingroup$ @DanieleA thanks! I guess I was getting too lazy to check things by myself $\endgroup$
    – TCiur
    Commented Dec 13, 2021 at 12:19

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