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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
 MSC Classifications: 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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