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

TL;DR

This paper explores the Schur-finiteness and Bass-finiteness conjectures for quadric fibrations and sextic du Val del Pezzo surfaces, establishing their relations to the base variety and proving them in low-dimensional cases.

Contribution

It introduces a relation between the conjectures for fibrations and their bases using noncommutative motives, and proves these conjectures for certain low-dimensional cases and complete intersections.

Findings

Proves Schur-finiteness for low-dimensional bases.

Establishes a relation between conjectures for fibrations and bases.

Proves Bass-finiteness conjecture in similar contexts.

Abstract

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.