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Birational geometry of quadrics. (English) Zbl 1221.14014

The author studies two important conjectures in the algebro-geometric theory of quadratic forms, the ruledness conjecture and the quadratic Zariski problem. Let \(k\) be a field of characteristic not \(2\) and let \(\phi\) be a quadratic form over \(k\) (which we may assume to be of dimension \(n\geq 3\) and anisotropic as otherwise most results will hold for trivial reasons), with associated projective quadric \(X_\phi\). The function field of \(\phi\) is defined as \(k(\phi)=k(X_\phi)\), and \(\phi\) will be isotropic over \(k(\phi)\). The first Witt index of \(\phi\) is then defined to be the Witt index of \(\phi_{k(\phi)}\). \(X_\phi\) (or \(\phi\)) is said to be ruled if \(X_\phi\) is birational to \(Y\times\mathbb{P}^1\) for some variety \(Y\) over \(k\). A theorem by Karpenko and Merkurjev implies that if \(i_1(\phi)=1\) then \(X_\phi\) is not ruled. The ruledness conjecture states that if \(i_1(\phi)\geq 2\) then \(\phi\) is ruled. Earlier, the author proved the conjecture to be true provided \(\dim\phi\leq 9\). In the present paper, the veracity of the conjecture is shown for odd-dimensional forms of dimension \(\leq 17\) and even-dimensional forms of dimension \(\leq 10\) or \(=14\). As an important ingredient, the author determines the structure of \(14\)-dimensional forms with \(i_1\geq 2\). He includes an example due to Vishik of a \(16\)-dimensional form with \(i_1=2\) that falls outside the class of forms for which ruledness could be determined previously. It is shown that this form and certain generalizations of it are indeed ruled as expected.
Let now \(\psi\) be another anisotropic form over \(k\) with \(\dim\phi =n =\dim\psi\). The quadratic Zariski problem asks if \(X_\phi\) and \(X_\psi\) being stably birational (equivalently, if \(\phi_{F(\psi)}\) and \(\psi_{F(\phi)}\) being both isotropic) implies that \(X_\phi\) and \(X_\psi\) are birational. By results of Roussey, Ahmad-Ohm and others, this is known to be true for \(n\leq 7\). The author extends this result to the class of (generalized) Pfister neighbors. Here, a form \(\sigma\) is called a neighbor of an \(r\)-fold Pfister form \(\rho\), \(r\geq 1\), if \(\dim \sigma\geq 2\) and there exists a form \(\alpha\) such that \(\sigma\) is similar to a subform of \(\alpha\otimes\rho\) of codimension \(<\dim\rho=2^r\). \(\sigma\) is called special if it itself contains a subform of type \(\beta\otimes\rho\) with \(\dim\beta=\dim\alpha-1\). It is easy to see that neighbors of the same dimension of the same multiple of an \(r\)-fold Pfister form will be stably birational. S. Roussey [Isotropie, corps de fonctions et équivalences birationnelles des formes quadratiques. PhD thesis. Université de Franche-Comté (2005)] proved that two special neighbors of the same dimension of some \(r\)-fold Pfister form \(\sigma\), \(r\leq 3\), are in fact birational. The author extends this result to arbitrary neighbors (resp. special neighbors) of the same multiple of some \(r\)-fold Pfister form with \(1\leq r\leq 3\) (resp. with \(r\geq 1\)). As a corollary, one obtains that the ruledness conjecture and the quadratic Zariski problem have a positive answer for arbitrary Pfister neighbors (in the classical sense) of Pfister forms of fold \(\leq 4\). Recall that \(\phi\) and \(\psi\) of the same dimension \(2^r\) are called half-neighbors if \(a\phi\perp b\psi\) is an \((r+1)\)-fold Pfister form for some \(a,b\in k^*\). Again, one readily checks that half-neighbors are stably birational, but it is still open if half-neighbors are birational in general. The author applies his construction of birational maps to an example of nonsimilar \(8\)-dimensional half-neighbors (due to the reviewer) and shows that they are birational, the first example of its kind.

MSC:

14E05 Rational and birational maps
14C25 Algebraic cycles
11E04 Quadratic forms over general fields
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References:

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