A supersingular K3 surface in characteristic 2 and the Leech lattice

By Igor Dolgachev, Shigeyuki Kondo. 2001

74 citations recorded by the source corpus.

We construct a K3 surface over an algebraically closed field of characteristic 2 which contains two sets of 21 disjoint smooth rational curves such that each curve from one set intersects exactly 5 curves from the other set. This configuration is isomorphic to the configuration of points and lines on the projective plane over the finite field of 4 elements. The surface admits a finite automorphism group which contains the group of automorphisms of the plane which acts on the configuration of each set of 21 smooth rational curves, and the additional element of order 2 which interchanges the two sets. The Picard lattice of the surface is a reflective sublattice of an even unimodular lattice of signatuire (1,25) and the classes of the 42 curves correspond to some Leech roots in this lattice.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8634ae304db97ef8ccdefba9c90182dda5b7432c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Serge Cantat
2. J. Blanc
3. Igor Dolgachev
4. D. Huybrechts

In Geometry and Topology

1. Serge Cantat
2. J. Blanc
3. Igor Dolgachev
4. D. Huybrechts
5. Keiji Oguiso

In Mathematical Physics

1. Serge Cantat
2. J. Blanc
3. Igor Dolgachev
4. Keiji Oguiso

Citations in the corpus

• Reflection groups in algebraic geometry (2006, background): “ver an algebraically closed field of characteristic 2 isomorphic to a quartic surface in P3 with equation T4 0 +T 4 1 +T 4 2 +T 4 3 +T 2 0 T 2 1 +T 2 0 T 2 2 +T 2 1 T 2 2 +T0T1T2(T0 +T1 +T2) = 0. (see [34]). Note that over a field of positive characteristic H2(X,Z) does not make sense, and the inequal”
• Dynamics of automorphisms of compact complex surfaces (2014, background): “More interestingly, one can construct surfaces in positive characteristic with automorphisms groups which are surprisingly large compared to the case of complex surfaces (see the theory of complex multiplication for tori, [60] for an example on a K3 surface, and [31] for rational surfaces).”
• Dynamical degrees of birational transformations of projective surfaces (2013, methodology): “To construct such an example, we make use of one of the main results of [22].”