The dihedral group 𝒟₅ as a group of symplectic automorphisms on K3 surfaces

Garbagnati, Alice

doi:10.1090/S0002-9939-2011-10650-6

The dihedral group $\mathcal D_5$ as a group of symplectic automorphisms on K3 surfaces
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Proc. Amer. Math. Soc. 139 (2011), 2045-2055 Request permission

Abstract:

We prove that if a K3 surface $X$ admits $\mathbb {Z}/5\mathbb {Z}$ as a group of symplectic automorphisms, then it actually admits $\mathcal {D}_5$ as a group of symplectic automorphisms. The orthogonal complement to the $\mathcal {D}_5$-invariants in the second cohomology group of $X$ is a rank 16 lattice, $L$. It is known that $L$ does not depend on $X$: we prove that it is isometric to a lattice recently described by R. L. Griess Jr. and C. H. Lam. We also give an elementary construction of $L$.

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