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The Ample Cone for a K3 Surface

  Published:2011-02-25
 Printed: Jun 2011
  • Arthur Baragar,
    Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, U.S.A.
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Abstract

In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.
Keywords: Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics
MSC Classifications: 14J28, 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05 show english descriptions $K3$ surfaces and Enriques surfaces
Automorphisms of surfaces and higher-dimensional varieties
Higher degree equations; Fermat's equation
Equations in many variables [See also 11P55]
Automorphism groups of lattices
Abelian varieties of dimension $> 1$ [See also 14Kxx]
Conformal densities and Hausdorff dimension
Hyperbolic orbits and sets
14J28 - $K3$ surfaces and Enriques surfaces
14J50 - Automorphisms of surfaces and higher-dimensional varieties
11D41 - Higher degree equations; Fermat's equation
11D72 - Equations in many variables [See also 11P55]
11H56 - Automorphism groups of lattices
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
37F35 - Conformal densities and Hausdorff dimension
37D05 - Hyperbolic orbits and sets
 

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