Stably rational surfaces over a quasi-finite field
- Authors: Colliot-Thélène J.1
-
Affiliations:
- Université Paris-Sud, Département de Mathématiques
- Issue: Vol 83, No 3 (2019)
- Pages: 113-126
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133778
- DOI: https://doi.org/10.4213/im8761
- ID: 133778
Cite item
Abstract
Let $k$ be a field and $X$ a smooth, projective,stably $k$-rational surface. If $X$ is split by a cyclic extension(for example, if the field $k$ is finite or, more generally, quasi-finite),then the surface $X$ is $k$-rational.
About the authors
Jean-Louis Colliot-Thélène
Université Paris-Sud, Département de Mathématiques
Email: jlct@math.u-psud.fr
References
- J.-L. Colliot-Thelène, J.-J. Sansuc, “La $R$-equivalence sur les tores”, Ann. Sci. Ecole Norm. Sup. (4), 10:2 (1977), 175–229
- J.-L. Colliot-Thelène, J.-J. Sansuc, “La descente sur les varietes rationnelles. II”, Duke Math. J., 54:2 (1987), 375–492
- A. Grothendieck, “Le groupe de Brauer. III. Exemples et complements”, Dix exposes sur la cohomologie des schemas, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968, 88–188
- J.-L. Colliot-Thelène, “Birational invariants, purity and the Gersten conjecture”, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., 58, Part I, Amer. Math. Soc., Providence, RI, 1995, 1–64
- A. Beauville, J.-L. Colliot-Thelène, J.-J. Sansuc, P. Swinnerton-Dyer, “Varietes stablement rationnelles non rationnelles”, Ann. of Math. (2), 121:2 (1985), 283–318
- J.-P. Serre, Corps locaux, Publ. Inst. Math. Univ. Nancago, VIII, Actualites Sci. Indust., no. 1296, 2-ème ed., Hermann, Paris, 1968, 245 pp.
- A. Pirutka, “Varieties that are not stably rational, zero-cycles and unramified cohomology”, Algebraic geometry: Salt Lake City 2015, v. 2, Proc. Sympos. Pure Math., 97, Part 2, Amer. Math. Soc., Providence, RI, 2018, 459–483
- J.-L. Colliot-Thelène, D. Coray, “L'equivalence rationnelle sur les points fermes des surfaces rationnelles fibrees en coniques”, Compositio Math., 39:3 (1979), 301–332
- У. Фултон, Теория пересечений, Мир, М., 1989, 583 с.
- S. Gille, “Permutation modules and Chow motives of geometrically rational surfaces”, With appendix by J.-L. Colliot-Thelène, J. Algebra, 440 (2015), 443–463
- В. А. Исковских, “Минимальные модели рациональных поверхностей над произвольными полями”, Изв. АН СССР. Сер. матем., 43:1 (1979), 19–43
- J. Kollar, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996, viii+320 pp.
- Ю. И. Манин, Кубические формы, Наука, М., 1972, 304 с.
- В. А. Исковских, “Рациональные поверхности с пучком рациональных кривых и с положительным квадратом канонического класса”, Матем. сб., 83(125):1(9) (1970), 90–119
- J.-L. Colliot-Thelène, J.-J. Sansuc, “On the Chow groups of certain rational surfaces: a sequel to a paper of S. Bloch”, Duke Math. J., 48:2 (1981), 421–447
- A. Varilly-Alvarado, “Arithmetic of del Pezzo surfaces”, Birational geometry, rational curves, and arithmetic, Simons Symp., Springer, Cham, 2013, 293–319
- В. А. Исковских, “Бирациональные свойства поверхности степени 4 в $mathbf P^4_k$”, Матем. сб., 88(130):1(5) (1972), 31–37
- J. S. Frame, “The classes and representations of the groups of 27 lines and 28 bitangents”, Ann. Mat. Pura Appl. (4), 32 (1951), 83–119
- H. P. F. Swinnerton-Dyer, “The zeta function of a cubic surface over a finite field”, Proc. Cambridge Philos. Soc., 63 (1967), 55–71
- T. Urabe, “Calculation of Manin's invariant for Del Pezzo surfaces”, Math. Comp., 65:213 (1996), 247–258
- B. Banwait, F. Fite, D. Loughran, “Del Pezzo surfaces over finite fields and their Frobenius traces”, Math. Proc. Cambridge Philos. Soc., Publ. online 2018
- Shuijing Li, Rational points on Del Pezzo surface of degree $1$ and $2$, Thesis (Ph.D.), Rice Univ., 2010, 78 pp.