Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$.

There exists a compactification, the Satake compactification, which is minimal and has the property that $$\overline{\mathcal{A}}_g=\mathcal{A}_g\coprod\overline{\mathcal{A}}_{g-1}.$$

It's well known that for a space in $\mathcal{A}_{g-1}$, the projectivized normal cone of the boundary in the whole thing is the Kummer of the point.

What about the higher codimension strata? For instance, what is the projectivized normal cone at a point for the embedding $\overline{\mathcal{A}}_1\subset \overline{\mathcal{A}}_g$, or $\overline{\mathcal{A}}_2\subset \overline{\mathcal{A}}_g$?

Is there a good general method for computing these?

`backward quotation`

marks. See "How to write math" on the side`$\longrightarrow$`

– Sándor Kovács May 6 '11 at 16:26