On stable rationality of Fano threefolds and del Pezzo fibrations

Hassett, Brendan; Tschinkel, Yuri

doi:10.1515/crelle-2016-0058

Cited by 32 publications

(31 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Brauer group, and more generally, unramified cohomology groups, have been used most notably in the context of the Lüroth problem and in Noether's problem, see [A-M72], [Sa77], [Bogo87], [CTO], [CT95], [Bogo05], [Pey08]. The Chow group of 0-cycles has gained increased attention since the advent of the degeneration method due to Voisin [Voi15], and further developed by Colliot-Thélène and Pirutka [CT-P16], which unleased a torrent of breakthrough results proving the non stable rationality of many types of conic bundles over rational bases [HKT15], [A-O16], [ABBP16], [BB16], hypersurfaces of not too large degree in projective space [To16], [Sch18], and many other geometrically interesting classes of rationally connected varieties, e.g., [CT-P16], [HT16], [HPT16]. See [Pey16] for a survey of many of these results.…”

Section: Introductionmentioning

confidence: 99%

Universal Triviality of the Chow Group of 0-cycles and the Brauer Group

Auel

Bigazzi

Böhning

et al. 2019

International Mathematics Research Notices

View full text Add to dashboard Cite

We prove that a smooth proper universally CH0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p. Conference on Birational Geometry August 2017 during which part of this work was presented and several participants gave us useful feedback. Background on Brauer groupsAll schemes below will be assumed to be Noetherian and k will denote a field unless otherwise specified. For a scheme X, we will denote by Br(X) = H 2 et (X, G m ) 4, 5

show abstract

Section: Introductionmentioning

confidence: 99%

Universal Triviality of the Chow Group of 0-cycles and the Brauer Group

Auel

Bigazzi

Böhning

et al. 2019

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…The rationality of smooth del Pezzo fibrations of lower degree has been studied extensively, as a result a nearly complete solution to the problem has been obtained. See for rationality and for stable rationality of fibrations of degree 4. Rationality for degrees 1, 2, and 3 has been studied in with the techniques of birational rigidity.…”

Section: Introductionmentioning

confidence: 99%

Birational geometry of del Pezzo fibrations with terminal quotient singularities

Krylov

2018

J. London Math. Soc.

View full text Add to dashboard Cite

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well‐studied question, but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non‐rationality, of del Pezzo fibrations with simple non‐Gorenstein singularities satisfying the K2‐condition. We then apply this result to study embeddings of prefixPSL2false(7false) into the Cremona group.

show abstract

“…An integral variety X over a field k is stably rational if X × P m is rational, for some m. In recent years, failure of stable rationality has been established for many classes of smooth rationally connected projective complex varieties, see, for instance [1,3,4,7,15,16,22,23,24,25,26,27,28,29,30,33,34,36,37]. These results were obtained by the specialization method, introduced by C. Voisin [37] and developed in [16].…”

Section: Introductionmentioning

confidence: 99%

Stable rationality of quadric and cubic surface bundle fourfolds

Auel

Böhning

Pirutka

2018

European Journal of Mathematics

View full text Add to dashboard Cite

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in P 2 × P 3 is not stably rational. Via projections onto the two factors, X → P 2 is a cubic surface bundle and X → P 3 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 Xn → P 2 with discriminant curve Dn ⊂ P 2 of degree 2n, such that Xn has nontrivial unramified Brauer group and admits a universally CH0-trivial resolution.

show abstract

On stable rationality of Fano threefolds and del Pezzo fibrations

Abstract: AbstractWe prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

Cited by 32 publications

References 31 publications

Universal Triviality of the Chow Group of 0-cycles and the Brauer Group

Universal Triviality of the Chow Group of 0-cycles and the Brauer Group

Birational geometry of del Pezzo fibrations with terminal quotient singularities

Stable rationality of quadric and cubic surface bundle fourfolds

Contact Info

Product

Resources

About