Using sheaf theoretic methods, we define functors Lπ0 : DMeff(k) → D(HI≤0(k)) and LAlb : DMeff(k) → D(HI≤1(k)). The functor LAlb extends the one in [3] to nonnecessarily geometric motives. These functors are then used to define higher Néron-Severi groups and higher Albanese sheaves.

In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.

This paper concerns complex algebraic K3 surfaces with an automorphism which acts trivially on the Néron-Severi group. Complementing a result by Vorontsov and Kondō, we determine those K3 surfaces where the order of the automorphism is a 2-power and equals the rank of the transcendental lattice. We also study the arithmetic of these K3 surfaces and comment on mirror symmetry.

Let X be the variety obtained by the Weil transfer with respect to a quadratic separable field extension of a generalized Severi-Brauer variety. We study (and, in some cases, determine) the canonical dimension, incompressibility, and motivic indecomposability of X . We determine canonical 2-dimension of X (in the general case).

Let X be a Brauer Severi variety over a field k of characteristic not 2 and let D be a division algebra over k with a k-linear involution. We investigate Witt groups of certain hermitian forms over D ⊗k OX .

In the current paper we prove the irreducibility of Severi varieties on Hirzebruch surfaces in arbitrary characteristic. Our approach is of purely algebro-geometric nature, and it works in any characteristic. As a result, we obtain a deformation-theoretic proof of the irreducibility of moduli spaces Mg in positive characteristic, which does not involve reduction to characteristic zero.

The notion of geometric k-normality for curves is introduced in complete generality and is investigated in the case of nodal and cuspidal curves living on several types of surfaces. We discuss and suggest some applications of this notion to the study of Severi varieties of nodal curves on surfaces of general type and on P 2 .

Using sheaf theoretic methods, we define functors Lπ0 : DMeff(k) → D(HI≤0(k)) and LAlb : DMeff(k) → D(HI≤1(k)). The functor LAlb extends the one in [3] to nonnecessarily geometric motives. These functors are then used to define higher Néron-Severi groups and higher Albanese sheaves. The latter may be considered as an algebraic avatar of Deligne (co)homology.