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

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 Rémy, 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.

v2: slightly shortened and improved version; close to the published one

References in corpus (1)

• On non-liftable Calabi-Yau threefolds

Cited by in corpus (4)

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• Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic
• Non-log liftable log del Pezzo surfaces of rank one in characteristic five