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

Abstract

Let $Q\to B$ be a quadric fibration and $T\to B$ 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.