J. M. Gamboa

Automorphism Groups of Compact Bordered Klein Surfaces

Preliminary results.- Klein surfaces as orbit spaces of NEC groups.- Normal NEC subgroups of NEC groups.- Cyclic groups of automorphisms of compact Klein surfaces.- Klein surfaces with groups of automorphisms in prescribed families.- The automorphism group of compact Klein surfaces with one boundary component.- The automorphism group of hyperelliptic compact Klein surfaces with boundary.

The full automorphism groups of hyperelliptic Riemann surfaces

For every integer g≥2 we obtain the complete list of groups acting as the full automorphisms groups on hyperelliptic Riemann surfaces of genus g.

Symmetries of Riemann surfaces on which PSL(2, q) acts as a Hurwitz automorphism group

Let X be a compact Riemann surface and Aut(X) be its automorphism group. An automorphism of order 2 reversing the orientation is called a symmetry. The authors together with D. Singerman have been working on symmetries of Riemann surfaces in the last decade. In this paper, the symmetry type St(X) of X is defined as an unordered list of species of conjugacy classes of symmetries of X, and for a class of particular surfaces, St(X) is found. This class consists of Riemann surfaces on which PSL(2, q) acts as a Hurwitz group. An algorithm to calculate the symmetry type of this class is provided.

Polynomial and regular images of R n

We obtain new necessary conditions for ann-dimensional semialgebraic subset of ℝn to be a polynomial image of ℝn. Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers.

On compact Riemann surfaces with dihedral groups of automorphisms

We study compact Riemann surfaces of genus g 2 having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group DN. The question of extendability of the action of DN is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

In this work we define a semialgebraic set S Rn to be irreducible if the noetherian ring of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring : Substitution Theorem, Positivstellens¨atze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

On orderings in real surfaces

After describing explicitly all total orderings in the ring R[[x, y]], we prove that each ordering in the quotient field of the ring of germs of real analytic functions at an irreducible point O of a real analytic surface X is defined by a half-branch of the germ at O of some curve on X, which is analytic off the origin. Then follows an analogous result for real algebraic surfaces.

Topological Types Of P-Hyperelliptic Real Algebraic-Curves

Given a natural number p, a projective irreducible smooth algebraic curve V defined over R is called p-hyperelliptic if there exists a birational isomorphism of V, of order 2, such that V/ has genus p. This work is concerned with the existence of such curves according to their genus g and the number k of connected components of V(R). We prove that Harnack’s condition 1 k g is sufficient if V \ V (R) is connected. In case V \ V (R) non-connected, the following conditions 1 k g + 1 (g + k 1(2)), and either k = g + 1 − 2q for some q, 0 q p, or k 2p + 2 with = 1 for even p, = 2 for odd p, are necessary and sufficient for the existence of the curve.

Ordered fields with the dense orbits property

For a formally real field K let X(K) be the space of orderings. The automorphism group Aut(K) of K operates on X(K). K is said to have the ’dense orbits property’ (DOP) if for all 2 X(K) the orbit of is dense in X(K). The study of such fields, which was begun in D. W. Dubois and the second author [Contemp. Math. 8, 265-288 (1982; Zbl 0484.12003)], is continued here. The main question considered is the behavior of the DOP under purely transcendental and under algebraic field extensions.

On rings of semialgebraic functions

The authors study some properties of the ring of abstract semialgebraic functions over a constructible subset of the real spectrum of an excellent ring. To be more precise, let X be a constructible subset of the real spectrum of a ring A. The ring S(X) of abstract semialgebraic functions over X was introduced bz N. Schwartz [see Mem. Am. Math. Soc. 397 (1989; Zbl 0697.14015)], as a generalization of continuous functions with semialgebraic graph to the context of real spectra. Unfortunately the utility of this functions is not yet quite established. The main result of the paper states that if A is excellent, the Krull dimension of S(X) equals the dimension of X (defined as the maximum of the heights of the supports of points lying in X), which in turn, as J. M. Ruiz showed in C. R. Acad. Sci. Paris, S´er. I 302, 67-69 (1986; Zbl 0591.13017) coincides with its topological dimension. This was first shown by M. Carral and M. Coste [J. Pure Appl. Algebra 30, 227-235 (1983; Zbl 0525.14015)] for the particular case of X being a ‘true’ semialgebraic subset which is locally closed, and the result extends readily to abstract locally closed constructible sets. Then the authors use the compactness of the constructible topology of real spectra and the properties of excellent rings to reduce the general case to the locally closed one. The paper finishes by characterizing the finitely generated prime ideals of S(X), namely they are the ideals of the open constructible points of X whose closure in X is open of dimension 6= 1.