Rational surfaces over perfect fields (résumé anglais)

Letk be a perfect field of arbitrary characteristic. The main object of this paper is to establish some new objects associated with algebraic surfaces F defined overk which are invariants for birational transformations defined overk. There are two main applications. The first is that if K is any extension ofk of degree 2, then there are infinitely many birationally inequivalent rational surfaces defined overk which all become birationally equivalent to the plane over K. The second application is to a partial classification of the del Pezzo surfaces for birational equivalence overk. For our purposes a del Pezzo surface defined overk is a nonsingular rational surface with a very ample anticanonical system, so the nonsingular cubic surfaces are a special care. As we use the language of schemes, we have to prove some classical results in the new framework, notably some results of Enriques [7] on the classification of rational surfaces. In the last section we produce evidence for the conjecture that if the fieldk is quasialgebraically closed (in the sense of Lang [11]), then a rational surface defined overk always has a point on it defined overk. We shall now describe the contents of our paper in more detail.