Arthur Baragar, University of Nevada
“Apollonian packings and Enriques surfaces”
A hypersphere packing is an Apollonian packing if the spheres do not intersect except tangentially (externally), they fill the space up to a set of measure zero, and the packing includes a maximal configuration of mutually tangent hyperspheres. (There are other properties one should want, like ”lattice” based.) Apollonian hypersphere packings were thought to not exist in dimensions greater than three, owing to a Mathematical Review of one of David Boyd’s papers. It turns out, not only do they exist, but George Maxwell (Boyd’s colleague at UBC) described them in a follow up paper, but apparently was unaware that they are Apollonian. (They were among a large set of examples of sphere packings, most of which are not Apollonian.) Now, suppose X is an Enriques surface that contains a smooth rational curve but is otherwise generic. Let Λ be its Picard group modulo torsion. Then Λ is a lattice in a Lorentz space with intersection pairing of signature (1, 9), so naturally includes a model of nine dimensional hyperbolic space H9. Any smooth rational curve C on X satisfies C · C = −2, so represents a plane in H9, and hence a hypersphere in R8, thought of as the boundary of the Poincar ́e upper halfspace model of H9. In this way, the set of all smooth rational curves on X gives a configuration of hyperspheres in R8. That configuration is an Apollonian packing! Furthermore, it is related to the E8 lattice.