Structures on a K3 surface.

摘要

In the first part of this paper, we examine properties of K3 surfaces of the form &parl0;x2+1&parr0;&parl0;y2+1&parr0; &parl0;z2+1&parr0;+Axyz-2=0. We show the surface has Picard number q ≥ 12 by finding 12 curves whose equivalence classes are linearly independent. These curves have self intersection -2. We find the lattice representations of the single-coordinate swapping automorphisms in x, y, and z. We show that we have enough of the Lattice to make accurate predictions of polynomial degree growth under the automorphisms. We describe these automorphisms in terms of operations on elliptic curves.;In the second part of this paper, we look at curves whose shape is sketched by the orbit of a point under the composed automorphisms mentioned above. These curves were studied by Fields Medalist Kurt McMullen. One can prove these curves are non-algebraic through the use of intersection theory. We offer a simple counting argument that one such curve is not algebraic. We do this by counting points in Fp and comparing this to the Hasse-Weil upper bound for such curves.