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Izhboldin’s results on stably birational equivalence of quadrics. (English) Zbl 1045.11024

Tignol, Jean-Pierre (ed.), Geometric methods in the algebraic theory of quadratic forms. Proceedings of the summer school, Lens, France, June 2000. Berlin: Springer (ISBN 3-540-20728-7/pbk). Lect. Notes Math. 1835, 151-183 (2004).
In this article, the author proves the results which have been announced in a draft by O. T. Izhboldin shortly before his tragic death. This draft has been published separately in the present volume [Geometric methods in the algebraic theory of quadratic forms. Proceedings of the summer school, Lens, France, June 2000, Lect. Notes Math. 1835, 143–150 (2004; Zbl 1045.11023), see the preceding review]. The results concern mainly the questions of when an anisotropic quadratic form \(\phi\) over a field \(F\) of characteristic \(\neq 2\) becomes isotropic over the function field \(F(\psi)\) of another quadratic form \(\psi\), or when two anisotropic forms \(\phi\) and \(\psi\) are stably equivalent, i.e., each becomes isotropic over the function field of the other form (this translates into the two associated projective quadrics being stably birationally equivalent).
Karpenko uses methods which at the time had not been at the disposal of Izhboldin when he announced his results. Let \(X=X_\phi\) be the quadric associated to a quadratic form \(\phi\) of odd dimension \(n=2r+1\), and assume that \(\phi\) is completely split (i.e., its Witt index is \(r\)). Using the fact that \(X\) is cellular and thus by taking a filtration by closed subsets \(X^{(i)}\) of codimension \(i\) with \(X^{(i)}\setminus X^{(i+1)}\) affine, one obtains a set of generators of \(CH^n(X\times X)\) given by the classes \([X^{(i)}\times X^{(n-i)}]\), \(0\leq i\leq n\). The type of a correspondence \(\alpha\in CH^n(X\times X)\) is then the sequence of the coefficients modulo \(2\) when writing \(\alpha\) as a linear combination of the above \([X^{(i)}\times X^{(n-i)}]\). If \(\phi\) is not split and if \(\alpha\in CH^n(X\times X)\), then the type of \(\alpha\) is defined to be the type of \(\alpha_{\bar{F}}\) where \(\bar{F}\) is an algebraic closure of the base field \(F\).
Formally, a type is just a finite sequence of elements in \({\mathbb Z}/2{\mathbb Z}\), and it is said to be possible for a given quadratic form \(\phi\) if there exists some correspondence on \(X_\phi\) of that type. Various operations can be performed with these types, and whether a certain type is possible for a form has important consequences on its isotropy behaviour over field extensions and its higher Witt indices. In order to treat the case of stable equivalence of quadratic forms \(\phi\) and \(\psi\) of the same odd dimension, one can also define possible types for the pair \((\phi ,\psi)\) by working with \(CH^n(X_\phi\times X_\psi)\). Types can also be defined in the even-dimensional case, but a little more care has to be taken.
Using these elegant techniques, the author proves all results announced by Izhboldin and adds various new ones. For example, he gives a complete classification of forms which are stably equivalent to a given \(9\)-dimensional quadratic form (as announced by Izhboldin), and he does so also for \(7\)-dimensional forms (going beyond what was announced by Izhboldin). A complete solution in the \(8\)-dimensional case (except for one rather technical subcase) is also given.
For the entire collection see [Zbl 1034.14001].

MSC:

11E04 Quadratic forms over general fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
14C15 (Equivariant) Chow groups and rings; motives

Citations:

Zbl 1045.11023
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