Essential dimension of quadrics
The essential dimension \(\dim_{\text{es}}(X)\) of an anisotropic projective quadric over a field \(F\) of characteristic not 2 is defined to be \(\dim(X) - i(X) + 1\), where \(i(X)\) is the Witt index of \(X\) over the function field \(F(X)\) of \(X\). The main result of the paper states that if \(Y\) is another anisotropic quadric over \(F\) and \(Y\) is isotropic over \(F(X)\), then \(\dim_{\text{es}}(X) \leq \dim_{\text{es}}(Y)\) and the equality holds if and only if \(X\) is isotropic over \(F(Y)\). This confirms a conjecture of O. Izhboldin. Actually the authors prove a more general statement where \(Y\) is a complete variety with all closed points of even degree. Then, if \(Y\) has a closed point of odd degree over \(F(X)\), the same conclusions follow (with \(\dim_{\text{es}}(Y)\) replaced with \(\dim(Y)\)). A corollary to this more general result says that an anisotropic quadric \(X\) cannot be rationally compressed to a variety \(Y\) with all closed points of even degree and of dimension smaller than \(\dim_{\text{es}}(X)\). As an application of the main result the authors solve a problem raised in 1976 by M. Knebusch. They prove that the smallest transcendence degree of a generic splitting field of a quadric \(X\) equals \(\dim_{\text{es}}(X)\). Finally, another beautiful result strengthening the Arason-Pfister Hauptsatz is proved: if an anisotropic quadratic form \(\phi\) over \(F\) belongs to the \(n\)th power of the fundamental ideal of the Witt ring \(WF\) and its dimension is greater than \(2^n\), then the dimension is at least \(2^n + 2^{n-1}\). The result is ascribed to A. Vishik and the proof given in the paper to D. Hoffmann.
scientific article; zbMATH DE number 2001022