4
$\begingroup$

In the construction of Drinfeld space, one uses the following construction of formal schemes (see appendix B of "Period mappings and differential equations. From $\mathbb C$ to $\mathbb C_p$" by Yves Andre)

Let $R$ be a complete DVR with finite residue field $k$, $X$ is a smooth proper scheme over $R$. Then we blow up $X$ centered at all $k$-rational points of $X$ to get $X_1$, then we blow up $X_1$ centered at all $k$-rational points of $X$ to get $X_2$... The formal scheme is, roughly speaking, the limit of successive blow-ups $\dots \rightarrow X_2 \rightarrow X_1 \rightarrow X$.

What kind of formal scheme will we get for general $X$? For instance, what if $X$ is one integral model of a flag variety or abelian variety or high genus curve? What if we only assume $X$ is of semistable reduction?

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ In the case of surfaces, this idea appears in recent work of Oguiso on automorphism groups in positive characteristic. The point is that if $X$ and $Y$ are smooth proper surfaces over $k$ which are birational, then some $X_n$ admits a birational morphism to $Y$. $\endgroup$
    – Piotr Achinger
    Commented Oct 9, 2019 at 18:26

0

Reset to default

You must log in to answer this question.