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

Part of: Birational geometry Forms and linear algebraic groups Basic linear algebra

Published online by Cambridge University Press:  16 April 2018

Stephen Scully
Stephen Scully*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada (stephenjscully@gmail.com)
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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.

Keywords

quadratic formsisotropy indicesquadrics and quadratic Grassmanniansrational maps between quadratic Grassmannians

MSC classification

Primary: 11E04: Quadratic forms over general fields 14E05: Rational and birational maps 15A03: Vector spaces, linear dependence, rank
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 421 - 450
Copyright
© Cambridge University Press 2018

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