Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

3 Answers 3

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.

share|improve this answer
add comment

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 $A_g$.

share|improve this answer
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.