# Closure of image of Torelli map

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?

• The original reference is: W. Hoyt, On products and algebraic families of jacobian varieties. Ann. of Math. (2) 77 (1963), 415-423. – abx Jul 25 '17 at 5:46
• 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. – modnar Jul 25 '17 at 22:56
• see also Mumford's talk at Woods Hole, Further comments on boundariy points:alpha.math.uga.edu/%7Eroy/woodshole3.pdf – roy smith Jul 26 '17 at 4:14
• $\alpha (C,X)$ is defined in terms of $S$ on the first page of Hoyt's paper. – abx Jul 26 '17 at 5:54
• 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. – modnar Jul 26 '17 at 6:36