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Stable rationality - Lecture 1
Post-edited
Authors : Pirutka, Alena (Author of the conference)
CIRM (Publisher )
14C15 - (Equivariant) Chow groups and rings; motives
14C25 - Algebraic cycles
14H05 - Algebraic functions; function fields
14J70 - Algebraic hypersurfaces
14M20 - Rational and unirational varieties
14E08 - Rationality questions
CIRM (Publisher )
Abstract :
Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational nonrational varieties. This problem remained open till 1970th, when three types of such examples were produced: cubic threefolds (Clemens and Griffiths), some quartic threefolds (Iskovskikh and Manin), and some conic bundles (Artin et Mumford). The last examples are even not stably rational. The stable rationality of the first two examples was not known.
In a recent work C. Voisin established that a double solid ramified along a very general quartic is not stably rational. Inspired by this work, we showed that many quartic solids are not stably rational (joint work with J.-L. Colliot-Thélène). More generally, B. Totaro showed that a very general hypersurface of degree d is not stably rational if d/2 is at least the smallest integer not smaller than (n+2)/3. The same method allowed us to show that the rationality is not a deformation invariant (joint with B. Hassett and Y. Tschinkel).
In this series of lectures, we will discuss the methods to obtain the results above: the universal properties of the Chow group of zero-cycles, the decomposition of the diagonal, and the specialization arguments.
MSC Codes : In a recent work C. Voisin established that a double solid ramified along a very general quartic is not stably rational. Inspired by this work, we showed that many quartic solids are not stably rational (joint work with J.-L. Colliot-Thélène). More generally, B. Totaro showed that a very general hypersurface of degree d is not stably rational if d/2 is at least the smallest integer not smaller than (n+2)/3. The same method allowed us to show that the rationality is not a deformation invariant (joint with B. Hassett and Y. Tschinkel).
In this series of lectures, we will discuss the methods to obtain the results above: the universal properties of the Chow group of zero-cycles, the decomposition of the diagonal, and the specialization arguments.
14C15 - (Equivariant) Chow groups and rings; motives
14C25 - Algebraic cycles
14H05 - Algebraic functions; function fields
14J70 - Algebraic hypersurfaces
14M20 - Rational and unirational varieties
14E08 - Rationality questions
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 02/02/17 Conference Date : 25/01/17 Subseries : Research talks arXiv category : Algebraic Geometry Mathematical Area(s) : Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Video Time : 00:46:24 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2017-01-25_Pirutka_Part1.mp4 
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Information on the Event
Event Title : Algebraic geometry and complex geometry / Géométrie algébrique et géométrie complexeEvent Organizers : Broustet, Amaël ; Pasquier, Boris Dates : 23/01/17 - 27/01/17 Event Year : 2017 Event URL : http://conferences.cirm-math.fr/1593.html
Citation Data
DOI : 10.24350/CIRM.V.19114903Cite this video as: Pirutka, Alena (2017). Stable rationality - Lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19114903 URI : http://dx.doi.org/10.24350/CIRM.V.19114903 |
See Also
- [Multi angle] Liouville's inequality for transcendental points on projective varieties / Author of the conference Gasbarri, Carlo.
- [Multi angle] On the unirationality of Hurwitz spaces / Author of the conference Tanturri, Fabio.
- [Multi angle] Stable rationality - Lecture 3 / Author of the conference Pirutka, Alena.
- [Multi angle] Stable rationality - Lecture 2 / Author of the conference Pirutka, Alena.
Bibliography
- Beauville, A. (2016). The Lüroth problem. In R. Pardini, & G.P. Pirola (Eds.), Rationality problems in algebraic geometry (pp. 1-27). Cham: Springer - http://math1.unice.fr/%7Ebeauvill/conf/Cime.pdf
- Pirutka, A. (2016). Varieties that are not stably rational, zero-cycles and unramified cohomology.
- Voisin, C. (2016). Stable birational invariants and the Lüroth problem. In H.-D. Cao, & S.-T. Yau (Eds.), Advances in geometry and mathematical physics (pp. 313-342). Somerville, MA: International Press - https://webusers.imj-prg.fr/%7Eclaire.voisin/Articlesweb/jdgvoisin.pdf
