Witt groups of real projective surfaces
Let X be a smooth, integral, projective surface over \({\mathbb{R}}\). In this paper, we determine the structure of the Witt group W(X) of X in terms of certain birational invariants of X and compute W(X) explicitly for a projective, real, rational surface X (i.e. X with \(X_{{\mathbb{C}}}=X\times_{Spec({\mathbb{R}})}Spec({\mathbb{C}})\) birational to \({\mathbb{P}}^ 2({\mathbb{C}}))\). More precisely, if X is a projective, real, rational surface such that the space X(\({\mathbb{R}})\) of \({\mathbb{R}}\)-rational points of X is non-empty, we prove that \(W(X)\simeq {\mathbb{Z}}^ s\oplus ({\mathbb{Z}}/2)^{s-1}\), where s is the number of real connected components of X(\({\mathbb{R}})\), which is known to be finite and a birational invariant. The proof uses a result of J.-L. Colliot-Thélène and R. Parimala relating real components of algebraic varieties to unramified cohomology. If, on the other hand, \(X({\mathbb{R}})=\emptyset\), we prove that W(X)\(\simeq {\mathbb{Z}}/2\oplus {\mathbb{Z}}/4\). The proof of this uses a result of R. Parimala and the author, that for such an X, the level of the function field \({\mathbb{R}}(X)\) is 2. These results along with a theorem of Silhol, imply that a smooth, real, rational surface is birational to \({\mathbb{P}}^ 2({\mathbb{R}})\) if and only if W(X)\(\simeq {\mathbb{Z}}\).
birational characterization of real projective space
scientific article; zbMATH DE number 4132327