## projective subvarieties of the moduli space of abelian varieties

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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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.
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 $A_g$.