no. 3
Monodromy of the family of Cubic Surfaces branching over Smooth Cubic Curves
[Monodromie de la famille des surfaces cubiques se ramifiant sur des courbes cubiques lisses]
Medrano Martín del Campo, Adán1
1 Department of Mathematics The University of Chicago 5734 S. University Ave, Chicago, IL 60637 (USA)
Annales de l'Institut Fourier, Tome 72 (2022) no. 3, pp. 963-987.

On considère la famille des surfaces cubiques lisses qui sont réalisées comme recouvrements ramifieés triples de ℙ 2 , avec une courbe cubique lisse comme locus de ramification. Cette famille est paramétrée par l’espace 𝒰 3 des courbes cubiques lisses sur ℙ 2 et chaque surface est munie d’une action de groupe ℤ/3ℤ.

On calcule l’image de l’homomorphisme de monodromie ρ induit par l’action du π 1 (𝒰 3 ) sur les 27 droites contenues dans chaque surface cubique de cette famille. Grâce à un résultat classique, cette image est contenue dans le groupe de Weyl W(E 6 ). Notre résultat principal est que ρ est surjective sur le centralisateur de l’image d’un générateur du groupe. Notre démonstration est principalement calculatoire, et repose sur la relation entre les 9 points d’inflexion dans une courbe cubique et les 27 droites contenues dans la surface cubique se ramifiant dessus.

Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of ℙ 2 , with branch locus equal to a smooth cubic curve. This family is parametrized by the space 𝒰 3 of smooth cubic curves in ℙ 2 and each surface is equipped with a ℤ/3ℤ deck group action.

We compute the image of the monodromy map ρ induced by the action of π 1 (𝒰 3 ) on the 27 lines contained on the cubic surfaces of this family. Due to a classical result, this image is contained in the Weyl group WE 6 . Our main result is that ρ is surjective onto the centralizer of the image a of a generator of the deck group. Our proof is mainly computational, and relies on the relation between the 9 inflection points in a cubic curve and the 27 lines contained in the cubic surface branching over it.

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DOI : 10.5802/aif.3481
Classification : 14D05
Keywords: Monodromy, Cubic Surface, Cubic Curve
Mot clés : Monodromie, Surface Cubique, Courbe Cubique

Medrano Martín del Campo, Adán 1

1 Department of Mathematics The University of Chicago 5734 S. University Ave, Chicago, IL 60637 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Medrano Martín del Campo, Adán. Monodromy of the family of Cubic Surfaces branching over Smooth Cubic Curves. Annales de l'Institut Fourier, Tome 72 (2022) no. 3, pp. 963-987. doi : 10.5802/aif.3481. https://aif.centre-mersenne.org/articles/10.5802/aif.3481/

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