Birational geometry of quadrics
Bulletin de la Société Mathématique de France, Volume 137 (2009) no. 2, p. 253-276

We construct new birational maps between quadrics over a field. The maps apply to several types of quadratic forms, including Pfister neighbors, neighbors of multiples of a Pfister form, and half-neighbors. One application is to determine which quadrics over a field are ruled (that is, birational to the projective line times some variety) in a larger range of dimensions. We describe ruledness completely for quadratic forms of odd dimension at most 17, even dimension at most 10, or dimension 14. The proof uses a new structure theorem for 14-dimensional forms, generalizing Izhboldin's theorem on 10-dimensional forms. We also show that Vishik's 16-dimensional form is ruled.

Nous construisons de nouvelles applications birationnelles entre quadriques sur un corps. Diverses formes quadratiques sont considérées : les voisines de Pfister, les voisines des multiples d'une forme de Pfister, et les demi-voisines. Un corollaire est la détermination des quadriques réglées (c'est-à-dire birationnelles au produit de la droite projective et d'une variété) en certaines dimensions. Nous décrivons complètement les quadriques réglées lorsque la forme quadratique est de dimension impaire inférieure à 17, de dimension paire inférieure à 10, ou de dimension 14. La preuve utilise un nouveau théorème de structure sur les formes de dimension 14, généralisant le théorème d'Izhboldin sur les formes de dimension 10. Nous montrons également que la forme de Vishik de dimension 16 est réglée.

DOI : https://doi.org/10.24033/bsmf.2575
Classification:  11E04,  14E05
Keywords: quadratic forms, ruled varieties, birational geometry, quadratic Zariski problem
@article{BSMF_2009__137_2_253_0,
     author = {Totaro, Burt},
     title = {Birational geometry of quadrics},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {137},
     number = {2},
     year = {2009},
     pages = {253-276},
     doi = {10.24033/bsmf.2575},
     zbl = {1221.14014},
     mrnumber = {2543476},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2009__137_2_253_0}
}
Totaro, Burt. Birational geometry of quadrics. Bulletin de la Société Mathématique de France, Volume 137 (2009) no. 2, pp. 253-276. doi : 10.24033/bsmf.2575. http://www.numdam.org/item/BSMF_2009__137_2_253_0/

[1] H. Ahmad & J. Ohm - « Function fields of Pfister neighbors », J. Algebra 178 (1995), p. 653-664. | MR 1359908 | Zbl 0841.11015

[2] A. Borel - Linear algebraic groups, second éd., Graduate Texts in Math., vol. 126, Springer, 1991. | MR 1102012 | Zbl 0726.20030

[3] R. Elman, N. A. Karpenko & A. Merkurjev - The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, Amer. Math. Soc., 2008. | MR 2427530 | Zbl 1165.11042

[4] D. W. Hoffmann - « Isotropy of 5-dimensional quadratic forms over the function field of a quadric », in K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., 1995, p. 217-225. | MR 1327299 | Zbl 0824.11023

[5] -, « Similarity of quadratic forms and half-neighbors », J. Algebra 204 (1998), p. 255-280. | MR 1623969 | Zbl 0922.11031

[6] O. T. Izhboldin - « Quadratic forms with maximal splitting », Algebra i Analiz 9 (1997), p. 51-57. | MR 1468546 | Zbl 0892.11013

[7] -, « Fields of u-invariant 9 », Ann. of Math. 154 (2001), p. 529-587. | MR 1884616

[8] B. Kahn - « A descent problem for quadratic forms », Duke Math. J. 80 (1995), p. 139-155. | MR 1360614 | Zbl 0858.11024

[9] N. A. Karpenko - « Izhboldin's results on stably birational equivalence of quadrics », in Geometric methods in the algebraic theory of quadratic forms, Lecture Notes in Math., vol. 1835, Springer, 2004, p. 151-183. | MR 2066519 | Zbl 1045.11024

[10] N. A. Karpenko & A. Merkurjev - « Essential dimension of quadrics », Invent. Math. 153 (2003), p. 361-372. | MR 1992016 | Zbl 1032.11015

[11] M. Knebusch - « Generic splitting of quadratic forms. I », Proc. London Math. Soc. 33 (1976), p. 65-93. | MR 412101 | Zbl 0351.15016

[12] -, « Generic splitting of quadratic forms. II », Proc. London Math. Soc. 34 (1977), p. 1-31. | MR 427345 | Zbl 0359.15013

[13] M.-A. Knus, A. Merkurjev, M. Rost & J.-P. Tignol - The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, Amer. Math. Soc., 1998. | MR 1632779 | Zbl 0955.16001

[14] A. Laghribi - « Formes quadratiques de dimension 6 », Math. Nachr. 204 (1999), p. 125-135. | MR 1705142 | Zbl 0927.11023

[15] T. Y. Lam - Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, Amer. Math. Soc., 2005. | MR 2104929 | Zbl 1068.11023

[16] S. Roussey - « Isotropie, corps de fonctions et équivalences birationnelles des formes quadratiques », Thèse, Université de Franche-Comté, 2005.

[17] B. Totaro - « The automorphism group of an affine quadric », Math. Proc. Cambridge Philos. Soc. 143 (2007), p. 1-8. | MR 2340971 | Zbl 1124.14040

[18] A. Vishik - « Motives of quadrics with applications to the theory of quadratic forms », in Geometric methods in the algebraic theory of quadratic forms, Lecture Notes in Math., vol. 1835, Springer, 2004, p. 25-101. | MR 2066515 | Zbl 1047.11033

[19] A. R. Wadsworth & D. B. Shapiro - « On multiples of round and Pfister forms », Math. Z. 157 (1977), p. 53-62. | MR 506032 | Zbl 0355.15022