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?



projective subvarieties of the moduli space of abelian varieties
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.


