  Let Q -> B be a quadric fibration and T -> B a family of sextic du Val del
Pezzo surfaces. Making use of the recent 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.
