Book contents
- Frontmatter
- Dedicaton
- Contents
- Contributors
- Happy Birthday
- On Stable Cohomology of Central Extensions of Elementary Abelian Groups
- On Projective 3-Folds of General Type with pg = 2
- 15-Nodal Quartic Surfaces. Part I: Quintic del Pezzo Surfaces and Congruences of Lines in P3
- Mori Flips, Cluster Algebras and Diptych Varieties Without Unprojection
- The Mirror of the Cubic Surface
- Semi-Orthogonal Decomposition of a Derived Category of a 3-Fold With an Ordinary Double Point
- Duality and Normalization, Variations on a Theme of Serre and Reid
- Rationality of Q-Fano Threefolds of Large Fano Index
- An Exceptional Locus in the Perfect Compactification of Ag
- Variation of Stable Birational Types of Hypersurfaces
- Triangle Varieties and Surface Decomposition of Hyper-Käahler Manifolds
- References
Variation of Stable Birational Types of Hypersurfaces
Published online by Cambridge University Press: 25 October 2022
Book contents
- Frontmatter
- Dedicaton
- Contents
- Contributors
- Happy Birthday
- On Stable Cohomology of Central Extensions of Elementary Abelian Groups
- On Projective 3-Folds of General Type with pg = 2
- 15-Nodal Quartic Surfaces. Part I: Quintic del Pezzo Surfaces and Congruences of Lines in P3
- Mori Flips, Cluster Algebras and Diptych Varieties Without Unprojection
- The Mirror of the Cubic Surface
- Semi-Orthogonal Decomposition of a Derived Category of a 3-Fold With an Ordinary Double Point
- Duality and Normalization, Variations on a Theme of Serre and Reid
- Rationality of Q-Fano Threefolds of Large Fano Index
- An Exceptional Locus in the Perfect Compactification of Ag
- Variation of Stable Birational Types of Hypersurfaces
- Triangle Varieties and Surface Decomposition of Hyper-Käahler Manifolds
- References
Summary
We introduce and study the question how can stable birational types vary in a smooth proper family. Our starting point is the specialization for stable birational types of Nicaise and the author, and our emphasis is on stable birational types of hypersurfaces. Building up on the work of Totaro and Schreieder on stable irrationality of hypersurfaces of high degree, we show that smooth Fano hypersurfaces of large degree over a field of characteristic zero are in general not stably birational to each other. In the appendix, Claire Voisin proves a similar result in a different setting using the Chow decomposition of diagonal and unramified cohomology.
- Type
- Chapter
- Information
- Recent Developments in Algebraic GeometryTo Miles Reid for his 70th Birthday, pp. 296 - 313Publisher: Cambridge University PressPrint publication year: 2022
References
Artin, M. and D. Mumford. Some elementary examples of unirational varieties which are not rational. Proc. London Math. Soc. (3), 25:75–95, 1972.Google Scholar
Beauville, Arnaud. A very general sextic double solid is not stably rational. Bull. Lond.. Math. Soc., 48(2):321–324, 2016.Google Scholar
Bloch, Spencer and Arthur Ogus. Gersten's conjecture and the homology of schemes. Ann. Sci. Ecole Norm. Sup. (4), 7:181–201 (1975), 1974.CrossRefGoogle Scholar
Colliot-Thelene, Jean-Louis and Manuel Ojanguren. Varietes unirationnelles non rationnelles: au-dela de l'exemple d'Artin et Mumford. Invent. Math, 97(1):141–158, 1989.Google Scholar
Colliot-Thelene, Jean-Louis and Alena Pirutka. Cyclic covers that are not stably rational. Izv. Ross. Akad. Nauk Ser. Mat., 80(4):665–677, 2016.CrossRefGoogle Scholar
Colliot-Thelene, Jean-Louis and Alena Pirutka. Hypersurfaces quartiques de dimension 3: non-rationalite stable. Ann. Sci. Ec. Norm. Super. (4), 49(2):371–397, 2016.Google Scholar
Colliot-Thelene, Jean-Louis and Claire Voisin. Cohomologie non ramifiee et conjecture de Hodge entiere. Duke Math. J., 161(5):735–801, 2012.CrossRefGoogle Scholar
Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, William, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry.Google Scholar
Kontsevich, Maxim and Yuri Tschinkel. Specialization of birational types. Invent. Math., 217(2):415–432, 2019.Google Scholar
Nicaise, Johannes and Evgeny Shinder. The motivic nearby fiber and degeneration of stable rationality. Invent. Math., 217(2):377–413, 2019.CrossRefGoogle Scholar
Schreieder. On the rationality problem for quadric bundles. Duke Math. J., Stefan, 168(2):187–223, 2019.Google Scholar
Schreieder, Stefan. Stably irrational hypersurfaces of small slopes. J. Amer. Math.. Soc., 32(4):1171–1199, 2019.CrossRefGoogle Scholar
Schreieder. Variation of stable birational types in positive characteristic. arXiv:1903.06143, Stefan, 2019.CrossRefGoogle Scholar
Totaro, Burt. Hypersurfaces that are not stably rational. J. Amer. Math.. Soc., 29(3):883–891, 2016.Google Scholar
Voisin, Claire. Unirational threefolds with no universal codimension 2 cycle. Invent. Math, 201(1):207–237, 2015.CrossRefGoogle Scholar