Gamboa Mutuberria, José ManuelBujalance, E.Cirre, F.J.2023-06-202023-06-202008N.L. Alling, Real Elliptic Curves, in: Mathematical Studies, vol. 54, North-Holland, 1981. N.L. Alling, N. Greenleaf, Foundations of the Theory of Klein Surfaces, in: Lecture Notes in Math., vol. 219, Springer, 1971. E. Ballico, C. Keem, On multiple covering of irrational curves, Arch. Math. Soc. 65 (2) (1995) 151–160. E. Ballico, C. Keem, On double coverings of hyperelliptic curves, J. Pure Appl. Algebra 207 (2) (2006) 397–415. E. Bujalance, A classification of unbranched double coverings of hyperelliptic Riemann surfaces, Arch. Math. 47 (1) (1986) 93–96. E. Bujalance, J.A. Bujalance, G. Gromadzki, E. Martinez, The groups of automorphisms of nonorientable hyperelliptic Klein surfaces without boundary, in: Proc. of Groups-Korea (Pusan, 1988), in: Lecture Notes in Math., vol. 1398, Springer-Verlag, Berlin, Heidelberg, 1989, pp. 43–51. 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, Automorphism Groups of Compact Bordered Klein Surfaces, in: Lecture Notes in Math., vol. 1439, Springer-Verlag, Berlin, Heidelberg, 1990. F.J. Cirre, Birational classification of hyperelliptic real algebraic curves, in: The Geometry of Riemann Surfaces and Abelian Varieties, Contemp. 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. Gonz´alez-Diez, Smooth double coverings of hyperelliptic curves, in: The Geometry of Riemann Surfaces and Abelian Varieties, Contemp. Math. 397 (2006) 73–77. 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 22 (2) (1971) 117–123. H.H. Martens, A remark on Abel’s Theorem and the mapping of linear series, Comment. Math. Helvetici 52 (1977) 557–559.0022-404910.1016/j.jpaa.2007.12.008https://hdl.handle.net/20.500.14352/49927We consider double and (possibly) branched coverings : X ! X0 between real algebraic curves where X is hyperelliptic. 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. We first analyse the topological features and ramification data of such coverings. Second, for each isomorphism class of these coverings we then describe a representative, with defining polynomial equations for X and for X0, a formula for the involution that generates the coveri transformation group, and a rational formula forng the covering projection : X ! X0.engjournal articlehttp://www.sciencedirect.com/science/article/pii/S0022404908000182http://www.sciencedirect.comrestricted access512.772Real Algebraic CurvesHyperellipticRiemann SurfaceGeometria algebraica1201.01 Geometría Algebraica