Birational geometry of compactifications of Drinfeld half-spaces over a finite field

TL;DR

This paper explores the birational geometry of compactified Drinfeld half-spaces over finite fields, introducing new inseparable endomorphisms, studying foliations, and providing examples of special varieties in positive characteristic.

Contribution

It constructs a purely inseparable endomorphism of Drinfeld's half-space that does not extend to projective space and investigates foliations, leading to new examples of varieties in positive characteristic.

Findings

Constructed a purely inseparable endomorphism of Drinfeld's half-space.

Discovered a non-liftable Calabi-Yau threefold in characteristic 2.

Provided examples of rational surfaces with klt singularities and ample cotangent bundle.

Abstract

We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfeld's half-space $\Omega (V)$ over a finite field $k$ that does not extend to an endomorphism of the projective space $P (V)$. This should be compared with theorem of R\'emy, Thuillier and Werner that every $k$-automorphism of $\Omega (V)$ extends to a $k$-automorphism of $P (V)$. Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfeld's half-spaces. This leads to various examples of interesting varieties in positive characteristic. In particular, we show a new example of a non-liftable projective Calabi-Yau threefold in characteristic $2$ and we show examples of rational surfaces with klt singularities, whose cotangent bundle contains an ample line bundle.