hessian - TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY...

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 8, Pages 3031–3057 S 0002-9947(02)03011-8 Article electronically published on April 3, 2002 BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES IGOR DOLGACHEV AND JONGHAE KEUM 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 Kond¯o. 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. 0. Introduction Let S : F ( x 0 , x 1 , x 2 , x 3 ) = 0 be a nonsingular cubic surface in P 3 over C . Its Hessian surface is a quartic surface defined by the determinant of the matrix of second-order partial derivatives of the polynomial F . When F is general enough, the quartic H is irreducible and has 10 nodes. It contains also 10 lines which are the intersection lines of five planes in general linear position. The union of these five planes is classically known as the Sylvester pentahedron of S . The equation of S can be written as the sum of cubes of some linear forms defining the five planes. A nonsingular model of H is a K3-surface ˜ H . Its Picard number ρ satisfies the inequality ρ 16. In this paper we give an explicit description of the group Bir( H ) of birational isomorphisms of H when S is general enough so that ρ = 16. Although H , in general, does not have any non-trivial automorphisms (because S does not), the group Bir( H ) = Aut( ˜ H ) is infinite. We show that it is generated by the automorphisms defined by projections from the nodes of H , a birational involution which interchanges the nodes and the lines, and the inversion automorphisms of some elliptic pencils on ˜ H . This can be compared with the known structure of the group of automorphisms of the Kummer surface associated to the Jacobian abelian surface of a general curve of genus 2 (see [Ke2], [Ko]). The latter surface is birationally isomorphic to the Hessian H of a cubic surface ([Hu1]), but the Picard number of ˜ H is equal to 17 instead of 16. We use the method for computing Bir( H ) employed by Kond¯ o in [Ko]. We show that the Picard lattice S H of the K3-surface ˜ H can be primitively embedded into the unimodular lattice L = Λ U of signature Received by the editors June 30, 2001 and, in revised form, August 27, 2001. 2000 Mathematics Subject Classification. Primary 14J28, 14J50; Secondary 11H56.
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