[Les hypersurfaces de Coble]
Soit A une variété abélienne principalement polarisée indécomposable, de dimension g. Les fonctions thêta d'ordre 3 plongent A dans un espace projectif de dimension 3g−1, tandis que les fonctions thêta d'ordre 2 plongent la variété de Kummer X=A/{±1} dans un espace projectif de dimension 2g−1. Coble a observé que pour g=2 il existe une unique hypersurface cubique dans qui est singulière le long de A, et pour g=3 une unique hypersurface quartique dans singulière le long de X. Nous expliquons ces faits par une analyse élémentaire des représentations du groupe de Heisenberg correspondant.
Let A be an indecomposable principally polarized abelian variety of dimension g. Third order theta functions embed A in a projective space of dimension 3g−1, while second order theta functions embed the Kummer variety X=A/{±1} in a projective space of dimension 2g−1. Coble observed that for g=2 there is a unique cubic hypersurface in that is singular along A, and for g=3 a unique quartic hypersurface in singular along X. We explain these facts by a simple analysis of the representations of the corresponding Heisenberg group.
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Arnaud Beauville 1
@article{CRMATH_2003__337_3_189_0,
author = {Arnaud Beauville},
title = {The {Coble} hypersurfaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {189--194},
publisher = {Elsevier},
volume = {337},
number = {3},
year = {2003},
doi = {10.1016/S1631-073X(03)00302-9},
language = {en},
}
Arnaud Beauville. The Coble hypersurfaces. Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 189-194. doi : 10.1016/S1631-073X(03)00302-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00302-9/
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