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The separating semigroup of a real curve
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 79-96.

Nous introduisons le semi-groupe séparant d’une courbe algébrique réelle séparante. Les éléments de ce semi-groupe gardent trace des degrés possibles des revêtements du cercle obtenus par restriction à la partie réelle de la courbe des morphismes séparants. Nous introduisons aussi le semi-groupe hyperbolique, composé des éléments du semi-groupe séparant provenant des morphismes qui sont la composition d’une projection linéaire et d’un plongement de la courbe dans un espace projectif.

Nous déterminons les deux groupes dans le cas des courbes maximales. Nous démontrons aussi que tout plongement d’une courbe réelle séparante de degré suffisamment grand est hyperbolique. En utilisant ces semi-groupes, nous montrons que le lieu hyperbolique d’une courbe plongée n’est en général pas connexe.

We introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restricting separating morphisms to the real part of the curve. We also introduce the hyperbolic semigroup which consists of elements of the separating semigroup arising from morphisms which are compositions of a linear projection with an embedding of the curve to some projective space.

We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve of dividing type to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected.

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DOI : https://doi.org/10.5802/afst.1624
Classification : 14P99,  14H50,  14H51
@article{AFST_2020_6_29_1_79_0,
     author = {Mario Kummer and Kristin Shaw},
     title = {The separating semigroup of a real curve},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {79--96},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {1},
     year = {2020},
     doi = {10.5802/afst.1624},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1624/}
}
Mario Kummer; Kristin Shaw. The separating semigroup of a real curve. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 79-96. doi : 10.5802/afst.1624. https://afst.centre-mersenne.org/articles/10.5802/afst.1624/

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