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5 - Normal Forms for Kummer Surfaces
- By Adrian Clingher, Andreas Malmendier
Edited by Ron Donagi, University of Pennsylvania, Tony Shaska, Oakland University, Michigan
Book:

Integrable Systems and Algebraic Geometry

Published online:

19 March 2020

Print publication:

02 March 2020, pp 119-174
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Summary

We determine normal forms for the Kummer surfaces associated with abelian surfaces of polarization of type (1, 1), (1, 2), (2, 2), (2, 4), and (1, 4). Explicit formulas for coordinates and moduli parameters in terms of theta functions of genus two are also given. The normal forms in question are closely connected to the generalized Riemann identities for Theta functions of Mumford's.

8 - The Rigid Body Dynamics in an Ideal Fluid: Clebsch Top and Kummer Surfaces
- By Jean-Pierre Françoise, Daisuke Tarama
Edited by Ron Donagi, University of Pennsylvania, Tony Shaska, Oakland University, Michigan
Book:

Integrable Systems and Algebraic Geometry

Published online:

19 March 2020

Print publication:

02 March 2020, pp 288-312
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Summary

This is an expository presentation of a completely integrable Hamiltonian system of Clebsch top under a special condition introduced by Weber. After a brief account of the geometric setting of the system, the structure of the Poisson commuting first integrals is discussed following the methods by Magri and Skrypnyk. Introducing supplementary coordinates, a geometric connection to Kummer surfaces, a typical class of K3 surfaces, is mentioned and also the system is linearized on the Jacobian of a hyperelliptic curve of genus two determined by the system. Further some special solutions contained in some vector subspace are discussed. Finally, an explicit computation of the action variables is introduced.

On K3 Surface Quotients of K3 or Abelian Surfaces
Alice Garbagnati
Journal:

Canadian Journal of Mathematics / Volume 69 / Issue 2 / 01 April 2017

Published online by Cambridge University Press:

20 November 2018, pp. 338-372

Print publication:

01 April 2017
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The aim of this paper is to prove that a $\text{K3}$ surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a $\text{K3}$ surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the $\text{K3}$ surfaces that are (rationally) $G$-covered by Abelian or $\text{K3}$ surfaces (in the latter case $G$ is an Abelian group). When $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases.
Moreover, we prove that a $\text{K3}$ surface ${{X}_{G}}$ is the minimal model of the quotient of an Abelian surface by a group $G$ if and only if a certain configuration of rational curves is present on ${{X}_{G}}$. Again, this result was known only in some special cases, in particular, if $G$ has order 2 or 3.