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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?

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  • $\begingroup$ Fixed LaTeX. For future reference: when something does not render, put the math part within backward quotation marks. See "How to write math" on the side $\longrightarrow$ $\endgroup$
    – Sándor Kovács
    Commented May 6, 2011 at 16:26
  • $\begingroup$ Thanks, I used to know that, but had forgotten $\endgroup$
    – Charles Siegel
    Commented May 6, 2011 at 16:40

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