Schur-Finiteness (and Bass-Finiteness) Conjecture for Quadric Fibrations and Families of Sextic du Val del Pezzo Surfaces

摘要

Let (Q o B) be a quadric fibration and (T o B) a family of sextic du Val del Pezzo surfaces. Making use of the theory of noncommutative mixed motives, we establish a precise relation between the Schur-finiteness conjecture for (Q), resp. for (T), and the Schur-finiteness conjecture for (B). As an application, we prove the Schur-finiteness conjecture for (Q), resp. for (T), when (B) is low-dimensional. Along the way, we obtain a proof of the Schur-finiteness conjecture for smooth complete intersections of two or three quadric hypersurfaces. Finally, we prove similar results for the Bass-finiteness conjecture.