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Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $10$) that is injective on $\mathbb C$ valued points.

On the other hand, consider the diagonal map $\Delta: \overline{\mathscr M}_{1,1}^4 \to \mathscr A_4$ whose image is one dimensional. Since the dimensions match up, the intersection of $\Delta$ with $\overline{\mathscr M_4}$ is a dimension zero scheme and corresponds to finitely many genus $4$ curves whose Jacobian is completely split into isomorphic elliptic curves (geometrically, over $\mathbb C$) - unless the diagonal is contained in the image of the Torelli locus!?

What is the intersection number here/ what is known about these finitely many genus $4$ curves.

I found this paper that gives $3$ examples of such genus $4$ curves but what else is known?

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  • $\begingroup$ Sorry, why is the image of $M_{1,1}^4$ one dimensional? It seems like it should be four dimensional. Or did you mean to ask only about genus four curves whose Jacobian is a fourth power? $\endgroup$
    – David E Speyer
    Commented Dec 9, 2019 at 21:21
  • $\begingroup$ Right, that is the diagonal map. Sorry! I should clarify. $\endgroup$
    – Asvin
    Commented Dec 9, 2019 at 21:22
  • $\begingroup$ Maybe you mean that the Jacobian is isogenous (not isomorphic) to the fourth power of an elliptic curve. Maybe this weaker condition alters finiteness. $\endgroup$
    – François Brunault
    Commented Dec 9, 2019 at 22:44
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    $\begingroup$ $\mathscr{A}_4$ is the moduli space of principally polarized abelian varieties of dimension $4$. Jacobians in $\mathscr{A}_4$ are irreducible, and therefore never isomorphic to a product (as p.p.a.v.'s). $\endgroup$
    – abx
    Commented Dec 10, 2019 at 5:38

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