The Ample Cone for a K3 Surface

Abstract

Abstract In this paper, we give several pictorial fractal representations of the ample or K¨ahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in ℙ1 × ℙ1 × ℙ1 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.