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?

On products and algebraic families of jacobian varieties. Ann. of Math. (2) 77 (1963), 415-423. $\endgroup$ – abx Jul 25 '17 at 5:46