15-Nodal quartic surfaces. Part II: the automorphism group
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15-Nodal quartic surfaces. Part II: the automorphism group Igor Dolgachev1 · Ichiro Shimada2 Received: 24 September 2019 / Accepted: 14 October 2019 / Published online: 31 October 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract We describe a set of generators and defining relations for the group of birational automorphisms of a general 15-nodal quartic surface in the complex projective 3-dimensional space. Keywords Quartic surface · K 3 surface · Automorphism group · Lattice Mathematics Subject Classification 14J50 · 14Q10 · 20F55
1 Introduction A quartic surface in P3 with 16 ordinary double points as its only singularities is classically known as a Kummer quartic surface, and has been intensively investigated since 19th century (see, for example, Hudson [9] or Baker [1]). For a positive integer n ≤ 16, we denote by X n a quartic surface in P3 with n ordinary double points (nodes) satisfying the following assumptions: (i) The quartic surface X n can be degenerated by acquiring additional 16 − n nodes and hence becomes isomorphic to a Kummer quartic surface. (ii) The Picard lattice Sn of the minimal resolution Yn of X n is embeddable into the Picard lattice of a general Kummer surface by specialization, and it is generated over Q by the class h 4 ∈ Sn of a plane section of X n and the classes of the exceptional curves of the resolution Yn → X n . In particular, the rank of Sn is n + 1. (iii) The only isometries of the transcendental lattice Tn of Yn that preserve the subspace H 2,0 (Yn ) of Tn ⊗ C are ±1.
The second author was supported by JSPS KAKENHI Grant Nos. 15H05738, 16H03926, and 16K13749.
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Ichiro Shimada [email protected] Igor Dolgachev [email protected]
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Department of Mathematics, University of Michigan, 2072 East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA
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Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan
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The group Aut(Y16 ) of birational automorphisms of a general Kummer quartic surface X 16 was described by Kondo [14]. We describe the automorphism group Aut(Y15 ) of the K 3 surface Y15 by the embedding of lattices S15 → S16 induced by the specialization of X 15 to X 16 . As was proved in [7], the 15-nodal quartic surfaces satisfy Condition (i) and form an irreducible family. If we choose a general member of this family, then Conditions (ii) and (iii) are satisfied. We give a generating set of Aut(Y15 ) explicitly in Theorem 5.9, and describe the defining relations of Aut(Y15 ) with respect to this generating set in Theorem 5.10. Our main tool is Borcherds’ method [2,3], which was also used in the calculation of Aut(Y16 ) by Kondo [14]. We also compute the defining relations from the tessellation of the nef-and-big cone calculated by Borcherds’ method. Our method is heavily computational, and is based on machine-aided calculations carried out by GAP [26]. Explicit numerical data is available from the second author’s
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