We study a new obstruction to the existence of integral and rational points for algebraic varieties over function fields, the differential descent obstruction. We prove that that is the only obstruction to the existence of integral points in affine varieties in characteristic zero and also, in most cases, for rational points on curves in arbitrary characteristic.

We present here new hierarchies of literal varieties of languages. Each language under consideration is a disjoint union of a certain collection of “basic” languages described here. Our classes of languages correspond to certain literal varieties of homomorphisms from free monoids onto nilpotent groups of class ≤ 2.

We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatoria...

We give a simpler and purely topological proof of Ernström’s class formula (1997) for the degree of dual varieties. Our new proof also allows us to obtain a formula describing the degrees of the associated varieties studied by Gelfand, Kapranov and Zelevinsky (1994).

We give elementary proofs, using suitable jet spaces, of some old and new structural results concerning finite-dimensional differential algebraic varieties (characteristic zero). We prove analogous results for difference algebraic varieties in characteristic zero. We also mention partial results and problems in the positive characteristic case.

G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kähler metric of constant scalar curvature. In [Don02] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson’s results and computations to a new case, that of reductive varieties.

We give elementary proofs, using suitable jet spaces, of some old and new structural results concerning nite-dimensional diierential algebraic varieties (characteristic zero). We prove analogous results for diierence algebraic varieties in characteristic zero. We also mention partial results and problems in the positive characteristic case.