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 Ω(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 Ω(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.