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A supersingular \(K3\) surface in characteristic \(2\) and the Leech lattice. (English) Zbl 1061.14031

Summary: We construct a \(K3\) surface in characteristic \(2\) as a 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 isomorphic to \(\text{PGL}(3,{\mathbb F}_{4}) \cdot 2\) such that a subgroup \(\text{PGL}(3,{\mathbb F}_{4})\) acts on the configuration of each set of \(21\) smooth rational curves, and the additional element of order \(2\) beinterchanges the two sets. The Picard lattice of the surface is a reflective sublattice of an even unimodular lattice \(\text{II}_{1,25}\) of signature \((1,25)\) and the classes of the \(42\) curves correspond to some Leech roots in \(\text{II}_{1,25}\).

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
51E20 Combinatorial structures in finite projective spaces
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
11H06 Lattices and convex bodies (number-theoretic aspects)
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