The Ample Cone for a K3 Surface

In this paper, we give several pictorial fractal representations of the ample or Kahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in P×P×P defined over a sufficiently large number field K, which 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. The ample cone or Kahler cone for a surface is a significant and often complicated geometric object. Though much is known about the ample cone, particularly for K3 surfaces, only a few non-trivial examples have been explicitly described. These include the ample cones with a finite number of sides (see [N1] for n = 3, and [N2, N3] for n ≥ 5; the case n = 4 is attributed to Vinberg in an unpublished work [N1]); the ample cone for a class of K3 surfaces with n = 3 [Ba3]; and the ample cones for several Kummer surfaces, which are K3 surfaces with n = 20 [V, K-K, Kon]. Though the complexity of the problem generically increases with n, the problem for K3 surfaces with maximal Picard number (n = 20) appear to be tractable because of the small size of the transcendental lattice. In this paper, we introduce accurate pictorial representations of the ample cone and the associated fractal for surfaces within a class of K3 surfaces with Picard number n = 4 (see Figures 1, 3, 4, and 5). As far as the author is aware, the associated fractal has not been studied in any great depth for any ample cone for which the fractal has a non-integer dimension, except the one in [Ba3]. The fractal in that case is Cantor-like (it is a subset of S) and rigorous bounds on its Hausdorff dimension are calculated in [Ba1]. The Hausdorff dimension of the fractal of this paper is estimated to be 1.296± .010. Our second main result is to relate the Hausdorff dimension of the fractal to the growth of the height of curves for an orbit of curves on a surface in this class. Precisely, let V be a surface within our class of K3 surfaces and let A = Aut(V/K) be its group of automorphisms over a sufficiently large number field K. Let D be an ample divisor on V and let C be a curve on V . Define NA(C)(t,D) = #{C′ ∈ A(C) : C′ ·D < t}. Here we have abused notation by letting C′ also represent the divisor class that contains C′. The intersection C′ ·D should be thought of as a logarithmic height of 2000 Mathematics Subject Classification. 14J28, 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05.