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A special configuration of $12$ conics and generalized Kummer surfaces

Abstract : A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the $16\mathbf{A}_{1}$-configurations for usual Kummer surfaces. In this paper we construct $9$ other such $9\mathbf{A}_{2}$-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $9\mathbf{A}_{2}$-configurations are obtained by taking the pullback of a certain configuration of $12$ conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of $X$ and of the generic fiber of its natural elliptic pencil.
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Preprints, Working Papers, ...
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Contributor : Xavier Roulleau Connect in order to contact the contributor
Submitted on : Tuesday, December 8, 2020 - 4:00:52 PM
Last modification on : Monday, December 28, 2020 - 3:54:02 PM

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  • HAL Id : hal-03046892, version 1
  • ARXIV : 2004.11421



David Kohel, Xavier Roulleau, Alessandra Sarti. A special configuration of $12$ conics and generalized Kummer surfaces. 2020. ⟨hal-03046892⟩



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