Birational automorphisms of quartic Hessian surfaces

Dolgachev, Igor; Keum, JongHae

doi:10.1090/S0002-9947-02-03011-8

Birational automorphisms of quartic Hessian surfaces
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by Igor Dolgachev and JongHae Keum

Trans. Amer. Math. Soc. 354 (2002), 3031-3057

DOI: https://doi.org/10.1090/S0002-9947-02-03011-8

Published electronically: April 3, 2002

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Abstract:

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice $II_{1,25}$ of rank 26 and signature $(1,25)$. The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondō. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.

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