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

##### Doc. Math. 25, 2339-2354 (2020)
DOI: 10.25537/dm.2020v25.2339-2354

### Summary

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.

### Mathematics Subject Classification

14A22, 14C15, 14D06

### Keywords/Phrases

Schur-finiteness conjecture, Bass-finiteness conjecture, quadric fibrations, du Val del Pezzo surfaces, noncommutative algebraic geometry, noncommutative mixed motives

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### Affiliation

Tabuada, Gonçalo
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK, and Departamento de Matemática and Centro de Matemática e Aplicações (CMA), FCT, UNL, Quinta da Torre, 2829-516 Caparica, Portugal