This volume contains the proceedings of a joint USA-USSR symposium on algebraic geometry, held in Chicago, USA, in June-July 1989.

Table of Contents

Theorems about good divisors on log fano varieties (case of index r>n-2).- Fano maps and fundamental groups.- Surjectivity of gaussian maps for line bundles of large degree on curves.- De Rham complex on toroidal variety.- On rank 2 vector bundles with c 1 2 =10 and c2=3 on Enriques surfaces.- Towards the problem of rationality of conic bundles.- On DG-modules over the de rham complex and the vanishing cycles functor.- More on computing invariants.- Effective methods in invariant theory.- On the structure of shafarevich-tate groups.- On the fundamental group of the complement of a hypersurface in ?n.- Braid group technique m complex geometry, II: From arrangements of lines and conics to cuspidal curves.- Notes on exceptional vector bundles and helices.- Hodge conjecture and mixed motives II.- Algebraic methods in the study of simple-elliptic singularities.- Singularity theory applied to ?-divisors.- A slight generalization of the mehta-ramanathan theorem.- Some properties of dual varieties and their applications in projective geometry.- Linear irreducible lie algebras and hodge structures.- Ussr participants.