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The Ample Cone for a K3 Surface

Published online by Cambridge University Press:  20 November 2018

Arthur Baragar*
Affiliation:
Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, U.S.A. email: baragar@unlv.nevada.edu
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Abstract

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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 ± .010.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2011

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