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In Lecture IV of Mumford's lecture notes ``Curves and their Jacobians'', it is stated that the closure of the image of the Torelli map sending a curve $C$ to its Jacobian is made up of products of Jacobians.

Is there a citation in Mumford's notes or a paper with a proof of this result? I haven't been able to find a reference/proof in Mumford's book or other sources but it seems like a ``standard'' result from the way it was cited here -- Is the Torelli map an immersion?

Edit: Thank you for the reference! Is there a way to relate this to compactifying $\mathcal{M}_g$ (which Mumford's notes seem to hint at)? More specifically: If a nonsingular curve $C$ ``approaches'' a singular curve $C_0$ corresponding to a point on $\overline{\mathcal{M}}_g\setminus \mathcal{M}_g$, is there something we can say about how their Jacobians are related/what happens to the Torelli map at the boundary?

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    $\begingroup$ The original reference is: W. Hoyt, On products and algebraic families of jacobian varieties. Ann. of Math. (2) 77 (1963), 415-423. $\endgroup$
    – abx
    Commented Jul 25, 2017 at 5:46
  • $\begingroup$ Thank you for the reference! Just to clarify, what is the definition of $\alpha(C, X)$ on the first page of the paper? I wasn't able to find a definition in the paper itself but found a reference to a paper of Matsusaka ([3] in the reference). However, that paper itself refers to Th. 1 and Th. 9 in "Variétés abéliennes et courbes algébriques" by Weil. Once I looked at this article, I couldn't find a reference to a function that looked similar except "$S(X)$" for some zero cycle $X$ on p. 111. $\endgroup$
    – modnar
    Commented Jul 25, 2017 at 22:56
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    $\begingroup$ see also Mumford's talk at Woods Hole, Further comments on boundariy points:alpha.math.uga.edu/%7Eroy/woodshole3.pdf $\endgroup$
    – roy smith
    Commented Jul 26, 2017 at 4:14
  • $\begingroup$ $\alpha (C,X)$ is defined in terms of $S$ on the first page of Hoyt's paper. $\endgroup$
    – abx
    Commented Jul 26, 2017 at 5:54
  • $\begingroup$ I think I was confused about whether this is the same as the $S$ in Hoyt's paper since $S$ was specifically defined for 0-cycles in part of Weil's article I was looking at and $Z$ is a cycle of arbitrary dimension on p. 2 of Matsusaka's paper "On a characterization of a Jacobian variety'". In fact, it is defined as $S(Z(u)) = \alpha(u) + c$ for some constant $c$, so it looks like $Z$ is ``evaluated'' somewhere but I'm still not sure what the definition of $Z(u)$ is. $\endgroup$
    – modnar
    Commented Jul 26, 2017 at 6:36

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