Stable rationality of quadric and cubic surface bundle fourfolds

Asher Auel; Christian Böhning; Alena Pirutka

doi:10.1007/s40879-018-0233-1

Stable rationality of quadric and cubic surface bundle fourfolds

Asher Auel, Christian Böhning, Alena Pirutka

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in [InlineEquation not available: see fulltext.] is not stably rational. Via projections onto the two factors, [InlineEquation not available: see fulltext.] is a cubic surface bundle and [InlineEquation not available: see fulltext.] is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any n⩾ 4 , new quadric surface bundle fourfolds [InlineEquation not available: see fulltext.] with discriminant curve [InlineEquation not available: see fulltext.] of degree 2n, such that X_n has nontrivial unramified Brauer group and admits a universally CH ₀-trivial resolution.

Original language	English (US)
Pages (from-to)	732-760
Number of pages	29
Journal	European Journal of Mathematics
Volume	4
Issue number	3
DOIs	https://doi.org/10.1007/s40879-018-0233-1
State	Published - Sep 1 2018

Keywords

Brauer group
Cubic surface bundles
Fano fourfolds
Quadric bundles
Stable rationality

ASJC Scopus subject areas

General Mathematics

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10.1007/s40879-018-0233-1

Cite this

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title = "Stable rationality of quadric and cubic surface bundle fourfolds",

abstract = "We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in [InlineEquation not available: see fulltext.] is not stably rational. Via projections onto the two factors, [InlineEquation not available: see fulltext.] is a cubic surface bundle and [InlineEquation not available: see fulltext.] is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any n⩾ 4 , new quadric surface bundle fourfolds [InlineEquation not available: see fulltext.] with discriminant curve [InlineEquation not available: see fulltext.] of degree 2n, such that Xn has nontrivial unramified Brauer group and admits a universally CH 0-trivial resolution.",

keywords = "Brauer group, Cubic surface bundles, Fano fourfolds, Quadric bundles, Stable rationality",

author = "Asher Auel and Christian B{\"o}hning and Alena Pirutka",

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doi = "10.1007/s40879-018-0233-1",

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T1 - Stable rationality of quadric and cubic surface bundle fourfolds

AU - Auel, Asher

AU - Böhning, Christian

AU - Pirutka, Alena

PY - 2018/9/1

Y1 - 2018/9/1

N2 - We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in [InlineEquation not available: see fulltext.] is not stably rational. Via projections onto the two factors, [InlineEquation not available: see fulltext.] is a cubic surface bundle and [InlineEquation not available: see fulltext.] is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any n⩾ 4 , new quadric surface bundle fourfolds [InlineEquation not available: see fulltext.] with discriminant curve [InlineEquation not available: see fulltext.] of degree 2n, such that Xn has nontrivial unramified Brauer group and admits a universally CH 0-trivial resolution.

AB - We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in [InlineEquation not available: see fulltext.] is not stably rational. Via projections onto the two factors, [InlineEquation not available: see fulltext.] is a cubic surface bundle and [InlineEquation not available: see fulltext.] is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any n⩾ 4 , new quadric surface bundle fourfolds [InlineEquation not available: see fulltext.] with discriminant curve [InlineEquation not available: see fulltext.] of degree 2n, such that Xn has nontrivial unramified Brauer group and admits a universally CH 0-trivial resolution.

KW - Brauer group

KW - Cubic surface bundles

KW - Fano fourfolds

KW - Quadric bundles

KW - Stable rationality

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JO - European Journal of Mathematics

JF - European Journal of Mathematics

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Stable rationality of quadric and cubic surface bundle fourfolds

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