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A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

  • Stephen Scully (a1)
Abstract

Let $q$ be an anisotropic quadratic form defined over a general field $F$ . In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$ , and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

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1. Elman, R., Karpenko, N. and Merkurjev, A., The Algebraic and Geometric Theory of Quadratic Forms, AMS Colloquium Publications, Volume 56 (American Mathematical Society, Providence, RI, 2008).
2. Fitzgerald, R. W., Function fields of quadratic forms, Math. Z. 178(1) (1981), 6376.
3. Grothendieck, A., Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes des schémas, Quatrième partie, Publ. Math. Inst. Hautes Études Sci. 32 (1967), 5361.
4. Haution, O., On the first Steenrod square for Chow groups, Amer. J. Math. 135 (2013), 5363.
5. Haution, O., Detection by regular schemes in degree two, Algebr. Geom. 2(1) (2015), 4461.
6. Hoffmann, D. W., Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220(3) (1995), 461476.
7. Hoffmann, D. W., Diagonal forms of degree p in characteristic p , in Algebraic and Arithmetic Theory of Quadratic Forms, Contemporary Mathematics, Volume 344, pp. 135183 (American Mathematical Society, Providence, RI, 2004).
8. Hoffmann, D. W. and Laghribi, A., Quadratic forms and Pfister neighbors in characteristic 2, Trans. Amer. Math. Soc. 356(10) (2004), 40194053.
9. Hoffmann, D. W. and Laghribi, A., Isotropy of quadratic forms over the function field of a quadric in characteristic 2, J. Algebra 295(2) (2006), 362386.
10. Izhboldin, O., Virtual Pfister neighbors and first Witt index. With an introduction by Nikita Karpenko, in Geometric Methods in the Algebraic Theory of Quadratic Forms, Lecture Notes in Mathematics, Volume 1835, pp. 131142 (Springer, Berlin, 2004).
11. Kahn, B., Formes quadratiques sur un corps, Cours Spécialisés, Volume 15 (Société Mathématique de France, Paris, 2008).
12. Karpenko, N. A., On the first Witt index of quadratic forms, Invent. Math. 153(2) (2003), 455462.
13. Karpenko, N. A., Canonical dimension, in Proceedings of the International Congress of Mathematicians, Volume II, pp. 146161 (Hindustan Book Agency, New Delhi, 2010).
14. Karpenko, N. A., Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties, J. Reine Angew. Math. 677 (2013), 179198.
15. Karpenko, N. and Merkurjev, A., Essential dimension of quadrics, Invent. Math. 153(2) (2003), 361372.
16. Knebusch, M., Generic splitting of quadratic forms. I, Proc. Lond. Math. Soc. (3) 33(1) (1976), 6593.
17. Knebusch, M., Generic splitting of quadratic forms. II, Proc. Lond. Math. Soc. (3) 34(1) (1977), 131.
18. Laghribi, A., Quasi-hyperbolicity of totally singular quadratic forms, in Algebraic and Arithmetic Theory of Quadratic Forms, Contemporary Mathematics, Volume 344, pp. 237248 (American Mathematical Society, Providence, RI, 2004).
19. Rost, M., Some new results on the Chow groups of quadrics. Preprint (1990).
20. Scully, S., Rational maps between quasilinear hypersurfaces, Compos. Math. 149(3) (2013), 333355.
21. Scully, S., On the splitting of quasilinear p-forms, J. Reine Angew. Math. 713 (2016), 4983.
22. Scully, S., Hoffmann’s conjecture for totally singular forms of prime degree, Algebra Number Theory 10(5) (2016), 10911132.
23. Scully, S., Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics, Preprint, 2017, arXiv:1609.07100v2, 18 pages.
24. Totaro, B., Birational geometry of quadrics in characteristic 2, J. Algebraic Geom. 17(3) (2008), 577597.
25. Vishik, A., Integral motives of quadrics, MPIM Preprint, 1998-13.
26. Vishik, A., Direct summands in the motives of quadrics, Preprint, 1999, https://www.maths.nottingham.ac.uk/personal/av/papers.html.
27. Vishik, A., Motives of quadrics with applications to the theory of quadratic forms, in Geometric Methods in the Algebraic Theory of Quadratic Forms, Lecture Notes in Mathematics, Volume 1835, pp. 25101 (Springer, Berlin, 2004).
28. Vishik, A., Excellent connections in the motives of quadrics, Ann. Sci. Éc. Norm. Supér. (4) 44(1) (2011), 183195.
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
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