ACTION OF THE AUTOMORPHISM GROUP ON THE ...
Document type :
Article dans une revue scientifique: Article original
Title :
ACTION OF THE AUTOMORPHISM GROUP ON THE JACOBIAN OF KLEIN'S QUARTIC CURVE II: INVARIANT THETA-FUNCTIONS
Author(s) :
Markushevich, Dimitri [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Moreau, Anne [Auteur]
Laboratoire de Mathématiques d'Orsay [LMO]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Moreau, Anne [Auteur]
Laboratoire de Mathématiques d'Orsay [LMO]
Journal title :
Épijournal de Géométrie Algébrique
Publisher :
EPIGA
Publication date :
2024
ISSN :
2491-6765
HAL domain(s) :
Mathématiques [math]/Géométrie algébrique [math.AG]
English abstract : [en]
Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman ...
Show more >Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the algebra of invariant theta-functions. Unlike the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling block.Show less >
Show more >Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the algebra of invariant theta-functions. Unlike the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling block.Show less >
Language :
Anglais
Peer reviewed article :
Oui
Audience :
Internationale
Popular science :
Non
ANR Project :
Comment :
23 pages, 1 figure. Introduction and abstract rewritten, references updated
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