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

Gamboa Mutuberria, José Manuel (1991) Compact Klein surfaces with boundary viewed as real compact smooth algebraic curves. Other. Real Academia de Ciencias Exactas, Físicas y Naturales.

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## 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.

Item Type: Monograph (Other) Real algebraic sets; Compact Riemann surfaces and uniformization Sciences > Mathematics > Algebra 17247 28 Nov 2012 09:53 28 Nov 2012 09:53

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