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Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$?

If the answer is not known, then what would one expect?

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Do you have a candidate map? –  Matt Jan 20 '13 at 0:38
No, but maybe something like Knudsen clutching can be used? –  Harry Jan 20 '13 at 10:40

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