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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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 
@article{AIF_2022__72_3_963_0,
     author = {Medrano Mart{\'\i}n del Campo, Ad\'an},
     title = {Monodromy of the family of {Cubic} {Surfaces} branching over {Smooth} {Cubic} {Curves}},
     journal = {Annales de l'Institut Fourier},
     pages = {963--987},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {3},
     year = {2022},
     doi = {10.5802/aif.3481},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3481/}
}
TY  - JOUR
AU  - Medrano Martín del Campo, Adán
TI  - Monodromy of the family of Cubic Surfaces branching over Smooth Cubic Curves
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 963
EP  - 987
VL  - 72
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3481/
DO  - 10.5802/aif.3481
LA  - en
ID  - AIF_2022__72_3_963_0
ER  - 
%0 Journal Article
%A Medrano Martín del Campo, Adán
%T Monodromy of the family of Cubic Surfaces branching over Smooth Cubic Curves
%J Annales de l'Institut Fourier
%D 2022
%P 963-987
%V 72
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3481/
%R 10.5802/aif.3481
%G en
%F AIF_2022__72_3_963_0
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/

[1] Dolgachev, Igor; Libgober, Anatoly S. On the fundamental group of the complement to a discriminant variety, Algebraic Geometry, Proc. Conf., Chicago Circle 1980 (Lecture Notes in Mathematics), Volume 862, Springer, 1981, pp. 1-25 | MR | Zbl

[2] Frame, J. Sutherland The classes and representations of the groups of 27 lines and 28 bitangents, Ann. Mat. Pura Appl., Volume 32 (1951) no. 83, pp. 83-119 | DOI | MR | Zbl

[3] Harris, Joe Galois Groups of Enumerative Problems, Duke Math. J., Volume 46 (1979) no. 4, pp. 715-719 | MR | Zbl

[4] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977, pp. 395-409 (Chapter V) | Zbl

[5] McMullen, Curtis T. Braid groups and Hodge theory, Math. Ann., Volume 355 (2013) no. 3, pp. 893-946 | DOI | MR | Zbl

[6] Morita, Shigeyuki Geometry of Characteristic Classes, Translations of Mathematical Monographs; Iwanami Series in Modern Mathematics, 199, American Mathematical Society, 2001 (Chapter 4) | DOI | Zbl

[7] Naik, Vipul Classification of groups of order 24, 2011 (https://groupprops.subwiki.org/wiki/classification_of_groups_of_order_24)

[8] The Sage Developers SageMath, the Sage Mathematics Software System (Version 9.2), 2021 (https://www.sagemath.org)

[9] Wolfram Research, Inc. Mathematica, Version 12.2 (Champaign, IL, 2020, https://www.wolfram.com/mathematica)

[10] Zariski, Oscar On the problem of existence of algebraic functions of two variables possessing a given branch curve, Am. J. Math., Volume 51 (1929), pp. 305-328 | DOI | MR | Zbl

Cité par Sources :