The dihedral group $\Dh_5$ as group of symplectic automorphisms on K3 surfaces

Alice Garbagnati

The dihedral group $\Dh_5$ as group of symplectic automorphisms on K3 surfaces

Algebraic Geometry 2010-07-08 v2

Authors: Alice Garbagnati

Abstract

We prove that if a K3 surface $X$ admits $\Z/5\Z$ as group of symplectic automorphisms, then it actually admits $\Dh_5$ as group of symplectic automorphisms. The orthogonal complement to the $\Dh_5$ -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 $L$ .

Keywords

K3 surfaces symplectic automorphisms dihedral group lattice cohomology group

Cite

@article{arxiv.0812.4518,
  title  = {The dihedral group $\Dh_5$ as group of symplectic automorphisms on K3 surfaces},
  author = {Alice Garbagnati},
  journal= {arXiv preprint arXiv:0812.4518},
  year   = {2010}
}

Comments

11 pages. Arguments revised, results unchanged. Final version, to appear in Proc. Amer. Math. Soc

The dihedral group $\Dh_5$ as group of symplectic automorphisms on K3 surfaces

Abstract

Keywords

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