Let k be a field and V be an algebraic fc-scheme of dimension w—1. V is called a Severi-Brauer /;-scheme if there exists a separable algebraic extension L/k such that V xkL and Pw_1(L)=Proj L[XX, • • • , Xn] are isomorphic as /.-schemes [7]. V is said to be split by L/k. V is called a trivial Severi-Brauer ^-scheme if V and -Pw_i(fc) are isomorphic as fcschemes. Let K/k be a finite Galois exten...

We use theorems of N. Karpenko about the incompressibility of Severi Brauer varieties and quadratic Weil transfers of Severi Brauer varieties to compute the minimal number of parameters needed to define conjugacy classes of involutions of given type on division algebras of 2-primary index. Similarly we compute the minimal number of parameters needed to define conjugacy classes of étale subalgeb...

In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them to describe projective models of Kummer surfaces of (1, d)polarized Abelian surfaces for d = 1, 2, 3. As a consequence we prove that in these cases the Néron–...

We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d , provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves that, in addition, satisfy tangency conditions of given orders wit...

For any central simple algebra, the Grothendieck Chow-motive of the corresponding Severi-Brauer variety is decomposed in a direct sum where each summand is a twisted motive of the Severi-Brauer variety corresponding to the underlying division algebra. It leads to decompositions in other theories (for instance, of K-cohomologies) because of the universal property of the Chow-motives. In the seco...

We consider the problem of constructing triangulations of projective planes over Hurwitz algebras with minimal numbers of vertices. We observe that the numbers of faces of each dimension must be equal to the dimensions of certain representations of the automorphism groups of the corresponding Severi varieties. We construct a complex involving these representations, which should be considered as...

We compute degrees of algebraic cycles on certain Severi-Brauer varieties and apply it to show that: a generic division algebra of index p and exponent p is not decomposable (in a tensor product of two algebras) for any prime p and any α except the case when p = 2 and 2 | α; the 2-codimensional Chow group CH of the Severi-Brauer variety corresponding to the generic division algebra of index 8 a...