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I know that the fibre of $A_{g,n}$ over $\mathbf{F}_p$ is quasi-projective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for its geometry?

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up vote 5 down vote accepted

The dimension is $g(g+1)/2$. The supersingular locus gives a large projective subvariety but I don't recall whether it is smooth or not. For references, look up the many papers of F. Oort.

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Over any field there are complete subvarieties of dimension $g-1$. Proof: the Satake compactification $A_g^S$ (which exists over $\mathbb Z$, by Faltings-Chai) has boundary of codimension $g$. So intersecting $A_g^S$, in some projective embedding, with an appropriate number of hyperplanes will give such subvarieties of $A_g$.

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Let me comment on Felipe Voloch's answer. The supersingular locus (i.e. the locus of $p$-rank zero) is indeed complete: it is a closed subvariety of $\overline A_g$ (choose a toroidal compactification) because the $p$-rank is lower semicontinuous, and it does not meet the boundary because a torus has positive $p$-rank. Moreover it will in fact have the largest possible dimension of a complete subvariety of $A_g$ (in positive characteristic); its dimension is $g(g-1)/2$. Indeed if there exists a complete subvariety of dimension $d$ and $\eta$ is an ample divisor class, then $\eta^d \neq 0$. Now $\lambda_1$ (1st Chern class of Hodge bundle) is ample and $\lambda_1^{1+g(g-1)/2}$ vanishes. For this see

van der Geer, Gerard. Cycles on the moduli space of abelian varieties. Moduli of curves and abelian varieties, 65--89, Aspects Math., E33, Vieweg, Braunschweig, 1999. MR1722539

In characteristic zero there is no complete subvariety of $A_g$ of this dimension. Hence the largest dimension of a compact subvariety depends on the characteristic!

Keel, Sean; Sadun, Lorenzo. Oort's conjecture for $A_ g\otimes\Bbb C$. J. Amer. Math. Soc. 16 (2003), no. 4, 887--900 (electronic). MR1992828

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