Let X be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron–Severi rank ρ(X) of X for the number of irreducible curves C on X with negative self-intersection and geometric genus less than b1(X)/4, where b1(X) is the first étale Betti number of X. The proof involves a hyperbolic analog of the theory of sphe...

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This pro...