Algebraic Geometry
Note on the monodromy conjecture for a space monomial curve with a plane semigroup
Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 177-187.

Roughly speaking, the monodromy conjecture for a singularity states that every pole of its motivic Igusa zeta function induces an eigenvalue of its monodromy. In this note, we determine both the motivic Igusa zeta function and the eigenvalues of monodromy for a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a plane branch. In particular, this yields a proof of the monodromy conjecture for such a curve.

En gros, la conjecture de la monodromie pour une singularité dit que chaque pôle de sa fonction zêta d’Igusa motivique induit une valeur propre de sa monodromie. Dans cette note, nous déterminons la fonction zêta d’Igusa motivique ainsi que les valeurs propres de la monodromie pour une courbe d’espace monomiale qui apparaît comme fibre spéciale d’une famille équisingulière dont la fibre générique est une branche plane. En particulier, il en résulte une démonstration de la conjecture de la monodromie pour une telle courbe.

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Accepted:
Published online:
DOI: 10.5802/crmath.30
Classification: 14E15, 14E18, 14H20, 14J17, 32S40
Martín-Morales, Jorge 1; Mourtada, Hussein 2; Veys, Willem 3; Vos, Lena 4

1 Centro Universitario de la Defensa, IUMA, Academia General Militar, Ctra. de Huesca s/n., 50090 Zaragoza, Spain
2 Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Batiment Sophie Germain, case 7012, 75205 Paris Cedex 13, France
3 KU Leuven, Departement Wiskunde, Celestijnenlaan 200B, bus 2400, 3001 Leuven, Belgium
4 KU Leuven, Departement wiskunde, Celestijnenlaan 200B, bus 2400, 3001 Leuven, Belgium
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Martín-Morales, Jorge; Mourtada, Hussein; Veys, Willem; Vos, Lena. Note on the monodromy conjecture for a space monomial curve with a plane semigroup. Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 177-187. doi : 10.5802/crmath.30. http://archive.numdam.org/articles/10.5802/crmath.30/

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