In this paper we study automorphisms g of order p of K3-surfaces defined over an algebraically closed field of characteristic p> 0. We divide all possible actions in the following cases according to the structure of the set of fixed points X g: X g is a finite set, X g contains a one-dimensional part D which is a positive divisor of Kodaira dimension κ(X,D) = 0,1,2. In the latter case we

Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

Abstract. We show that Mukai’s classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic p under the assumptions that (i) the order of the group is coprime to p and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without the assumption (ii) we classify all possible new groups which may appear. We prove that the assumption (i) on the order of the group is always satisfied if p> 11 and if p = 2, 3, 5,11 we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by p which are not contained in Mukai’s list. 1.

Abstract. This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3 surfaces with Picard number 20 which arise naturally in connection with the configuration. 1.

Abstract. I give various criteria for singularities to appear on geometric generic fibers of morphism between smooth schemes in positive characteristics. This involves local fundamental groups, jacobian ideals, projective dimension, tangent and cotangent sheaves, and the effect of Frobenius. As an application, I determine which rational double points do appear on geometric generic fibers.

Abstract. We investigate unibranched singularities of dual varieties of evendimensional smooth projective varieties in characteristic 2. 1.

(0.1) Let k denote a finite field with q = ps elements, let f ∈ k[x1,...,xn] be a polynomial and let Ψ: Fp → C ∗ be a non-trivial additive character. Consider the exponential sum: