Stable rationality of quadric and cubic surface bundle fourfolds

Asher Auel, Christian Böhning, Alena Pirutka

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)732-760
Number of pages29
JournalEuropean Journal of Mathematics
Volume4
Issue number3
DOIs
StatePublished - Sep 1 2018

Fingerprint

Cubic Surface
Quadric
Rationality
Bundle
Chow Groups
Brauer Group
Specialization
Discriminant
Hypersurface
Trivial
Projection
Cycle
Curve

Keywords

  • Brauer group
  • Cubic surface bundles
  • Fano fourfolds
  • Quadric bundles
  • Stable rationality

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Stable rationality of quadric and cubic surface bundle fourfolds. / Auel, Asher; Böhning, Christian; Pirutka, Alena.

In: European Journal of Mathematics, Vol. 4, No. 3, 01.09.2018, p. 732-760.

Research output: Contribution to journalArticle

Auel, Asher ; Böhning, Christian ; Pirutka, Alena. / Stable rationality of quadric and cubic surface bundle fourfolds. In: European Journal of Mathematics. 2018 ; Vol. 4, No. 3. pp. 732-760.
@article{b60f908794914dc680ba24edd11dc079,
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",
year = "2018",
month = "9",
day = "1",
doi = "10.1007/s40879-018-0233-1",
language = "English (US)",
volume = "4",
pages = "732--760",
journal = "European Journal of Mathematics",
issn = "2199-675X",
publisher = "Springer International Publishing AG",
number = "3",

}

TY - JOUR

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

UR - http://www.scopus.com/inward/record.url?scp=85053016889&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053016889&partnerID=8YFLogxK

U2 - 10.1007/s40879-018-0233-1

DO - 10.1007/s40879-018-0233-1

M3 - Article

VL - 4

SP - 732

EP - 760

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 3

ER -