no. 3
Monodromy of the family of Cubic Surfaces branching over Smooth Cubic Curves
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, Volume 72 (2022) no. 3, pp. 963-987.

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.

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.

Received:
Revised:
Accepted:
Published online:
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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, Volume 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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