Compact Klein surfaces with boundary viewed as real compact smooth algebraic curves

Abstract

This paper gives a very good systematic presentation of the equivalence between the algebraic function fields in one variable over the field $\bbfR$ of real numbers and the Klein surfaces. In section 1 Klein surfaces and morphisms between them are defined, and example as well as the basic facts about them are given. The double covering of a Klein surface and the quotient of a Riemann surface under an antianalytic involution is described, and it is noted that these two constructions are mutually inverse. Section 2 is devoted to the notion of a meromorphic function of a compact Klein surface. It is shown that the field of meromorphic functions of a compact Klein surface is an algebraic function field in one variable over $\bbfR$. Also there exists a functor of the category $\cal K$ of compact Klein surfaces to the category ${\cal F}\sb \bbfR$ of the algebraic function fields in one variable over $\bbfR$. An intensive study of the set $S(E\mid\bbfR)$ of proper valuation rings $V$ of $E\in{\cal F}\sb \bbfR$ with $V\supset\bbfR$ is the object of section 3. The main results of this section are:\par (a) The residue field of $V\in S(E\mid\bbfR)$ is $\bbfR$ iff $E$ admits some ordering with respect to which $V$ is convex.\par (b) The Riemann theorem about the dimension $\ell(L)$ of the space $L(D)$ associated to the divisor $D$ of $E\mid \bbfR$.\par With these notations it is proved in section 4 that $S(E\mid\bbfR)$ admits a unique structure of a Klein surface for which $p:S(E(\sqrt{- 1})(\bbfC)\to S(E\mid\bbfR)$ is a morphism of Klein surfaces and $M(S(E\mid\bbfR))=E$. Further it is shown that every compact Klein surface $S$ is isomorphic to $S(M(S)\mid\bbfR)$. Also: $S\mapsto M(S)$ and $E\mapsto S(E\mid\bbfR)$ give an equivalence between $\cal K$ and ${\cal F}\sb \bbfR$. Here the Klein surfaces with non empty boundary correspond to the formally real fields. Among a series of interesting comments and remarks we mention merely two:\par (i) There are several non homeomorphic curves with the same field of rational functions, but there is a unique one among them which is irreducible, compact, non-singular and affine.\par (ii) The Klein surface $S$ with empty boundary is orientable iff $M(S)$ contains $\bbfC$.\par This carefully written paper is very interesting and recommended even for specialists.