Publication: Normal coverings of hyperelliptic real algebraic curves
Loading...
Full text at PDC
Publication Date
2007
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
Abstract
We consider normal (possibly) branched, finite-sheeted coverings $ \pi:X\rightarrow X'$ between hyperelliptic real algebraic curves. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems in case $ X$ has the maximum number of ovals within its genus. We first analyze the topological features and ramification data of such coverings. For each isomorphism class we then describe a representative, with defining polynomial equations for $ X$ and for $ X'$, formulae for generators of the covering transformation group, and a rational formula for the covering $ \pi:X\rightarrow X'$.
Description
UCM subjects
Unesco subjects
Keywords
Citation
R. D. M. Accola, On lifting the hyperelliptic involution, Proc. Amer. Math. Soc., 122 (2), (1994), 341–347.
N. L. Alling, Real Elliptic Curves, Mathematical Studies 54, North-Holland, 1981.
N. L. Alling and N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math. 219, Springer, 1971
E. Bujalance, A classification of unbranched double coverings of hyperelliptic Riemann surfaces, Arch. Math. 47, (1) (1986), 93–96.
E. Bujalance, F. J. Cirre, J. M. Gamboa, Double coverings of hyperelliptic real algebraic curves, submitted.
E. Bujalance, J. J. Etayo, J. M. Gamboa, Hyperelliptic Klein surfaces, Quart. J. Math. Oxford, 36 (2) (1985), 141–157.
E. Bujalance, J. J. Etayo, J. M. Gamboa, G. Gromadzki, Automorphisms Groups of Compact Bordered Klein Surfaces, Lecture Notes in Math. 1439, Springer-Verlag, Berlin, Heidelberg, 1990.
F. J. Cirre, Birational classification of hyperelliptic real algebraic curves, The Geometry of Riemann Surfaces and Abelian Varieties, Contemporary Math. 397 (2006), 15–26.
H. M. Farkas, Unramified double coverings of hyperelliptic surfaces, J. Analyse Math. 20 (1976), 150–155.
H. M. Farkas, Unramified double coverings of hyperelliptic surfaces II, Proc. Amer. Math. Soc., 101 (3) (1987), 470–474
Y. Fuertes, G. González-Diez, Smooth double coverings of hyperelliptic curves. The Geometry of Riemann Surfaces and Abelian Varieties, Contemporary Math., 397 (2006), 73–77,
Y. Fuertes, G. González-Diez, On unramified normal coverings of hyperelliptic curves. J. of Pure and Applied Algebra, 208 (3) (2007), 1063–1070.
B. H. Gross, J. Harris, Real algebraic curves, Ann. Sci. ´Ecole Norm. Sup., 14 (1981), 157–182.
A. Harnack, Uber die Vieltheiligkeit der ebenen algebraischen Kurven, Mat. Ann., 10 (1876), 189–198.
R. Horiuchi, Normal coverings of hyperelliptic Riemann surfaces, J. Math. Kyoto Univ. 19 (3) (1979), 497–523.
E. Kani, Unramified double covers of hyperelliptic Klein surfaces, C. R. Math. Rep. Acad. Sci. Canada 9 (3) (1987), 133–138.
C. Maclachlan, Smooth coverings of hyperelliptic surfaces, Quart. J. Math. Oxford (2) 22 (1971), 117–123.
H. H. Martens, A remark on Abel’s Theorem and the mapping of linear series, Comment. Math. Helvetici 52 (1977), 557–559.