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Let Q -> B be a quadric fibration and T -> B a family of sextic du Val delPezzo surfaces. Making use of the recent theory of noncommutative mixedmotives, we establish a precise relation between the Schur-finitenessconjecture for Q, resp. for T, and the Schur-finiteness conjecture for B. As anapplication, we prove the Schur-finiteness conjecture for Q, resp. for T, whenB is low-dimensional. Along the way, we obtain a proof of the Schur-finitenessconjecture for smooth complete intersections of two or three quadrichypersurfaces. Finally, we prove similar results for the Bass-finitenessconjecture.

Schur-finiteness (and Bass-finiteness) conjecture for quadric fibrations and for families of sextic du Val del Pezzo surfaces