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

Gonçalo Tabuada

doi:10.25537/dm.2020v25.2339-2354

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

Gonçalo Tabuada

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Abstract

Let Q → B be a quadric fibration and T → 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.

Original language	English
Pages (from-to)	2339-2354
Number of pages	16
Journal	Documenta Mathematica
Volume	25
DOIs	https://doi.org/10.25537/dm.2020v25.2339-2354
Publication status	Published - 2020

Keywords

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

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10.25537/dm.2020v25.2339-2354Licence: CC BY

10012082000Final published version, 233 KBLicence: CC BY

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

abstract = "Let Q → B be a quadric fibration and T → 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.",

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

author = "Gon{\c c}alo Tabuada",

note = "Funding Information: info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT# The author is grateful to Joseph Ayoub for useful e-mail exchanges concerning the Schur-finiteness conjecture, and to the anonymous referee for her/his comments and for suggesting Remark 10. The author also would like to thank the Hausdorff Institute for Mathematics (HIM) in Bonn for its hospitality. ",

year = "2020",

doi = "10.25537/dm.2020v25.2339-2354",

language = "English",

volume = "25",

pages = "2339--2354",

journal = "Documenta Mathematica",

issn = "1431-0635",

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TY - JOUR

T1 - Schur-Finiteness (and Bass-Finiteness) Conjecture for Quadric Fibrations and Families of Sextic du Val del Pezzo Surfaces

AU - Tabuada, Gonçalo

N1 - Funding Information: info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT# The author is grateful to Joseph Ayoub for useful e-mail exchanges concerning the Schur-finiteness conjecture, and to the anonymous referee for her/his comments and for suggesting Remark 10. The author also would like to thank the Hausdorff Institute for Mathematics (HIM) in Bonn for its hospitality.

PY - 2020

Y1 - 2020

N2 - Let Q → B be a quadric fibration and T → 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.

AB - Let Q → B be a quadric fibration and T → 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.

KW - Bass-finiteness

KW - conjecture

KW - du Val del Pezzo surfaces

KW - noncommutative algebraic geometry

KW - noncommutative mixed motives

KW - quadric fibrations

KW - Schur-finiteness conjecture

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U2 - 10.25537/dm.2020v25.2339-2354

DO - 10.25537/dm.2020v25.2339-2354

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AN - SCOPUS:85099564782

SN - 1431-0635

VL - 25

SP - 2339

EP - 2354

JO - Documenta Mathematica

JF - Documenta Mathematica

ER -

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

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Keywords

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