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We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfelds 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 Remy, 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 Drinfelds 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.
We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category d
We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.
We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here random means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ c