On the Normal Bundle of Curves on Complete Intersection Surfaces.
We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br(S)/Br(ℚ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.
We introduce a new fundamental group scheme for varieties defined over an algebraically closed (or just perfect) field of positive characteristic and we use it to study generalization of C. Simpson’s results to positive characteristic. We also study the properties of this group and we prove Lefschetz type theorems.