A special configuration of 12 conics and generalized Kummer surfaces

Kohel, David; Roulleau, Xavier; Sarti, Alessandra

doi:10.1007/s00229-021-01334-2

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A special configuration of 12 conics and generalized Kummer surfaces

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Roulleau, Xavier
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s00229-021-01334-2
(Publisher version)

https://doi.org/10.48550/arXiv.2004.11421
(Preprint)

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MPIM Preprint series 2020-12.pdf
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Citation

Kohel, D., Roulleau, X., & Sarti, A. (2022). A special configuration of 12 conics and generalized Kummer surfaces. Manuscripta Mathematica, 169(3-4), 369-399. doi:10.1007/s00229-021-01334-2.

Cite as: https://hdl.handle.net/21.11116/0000-0009-255E-5

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.