Picard lattices of families of K3 surfaces.

Abstract

It is a nontrivial problem to determine the Picard Lattice of a given surface; the object of this thesis is to compute the Picard Lattices of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective space. Reid's list arises in many problems; here we look at an application to Mirror Symmetry. One can define an analogue of Mirror Symmetry for Calabi-Yau threefolds for K3 surfaces. This analogue coincides with the strange duality for the 14 surface singularities of V.I. Arnold. Here we investigate the Mirror Symmetry of Reid's 95 families of surfaces; Arnold's singularities are on this list. Denote a surface by S and a mirror family by S. Then we define a mirror family by$${\rm Pic}(S)\sbsp{H\sp2(S,\doubz)}{\perp} = {\rm Pic}(\check S)\perp U.$$By computing the Picard Lattice for each of these 95 surfaces, I am able to determine whether the mirror family for each one is also on Reid's list. I begin the thesis by reviewing the history of the problem, and then discuss the problems with computing the rank $\rho$ of the Picard Lattice. The bulk of Chapter 1 is devoted to exposition of lattices/quadratic forms and background on K3 elliptic surfaces. There is also an explanation of the computer programs I wrote to assist with the computations. Chapter 2 concerns a conjecture by Reid on how to compute the Picard Lattice. I discuss this conjecture and investigate the new lattices $M\sb{\vec p,\vec i,k}$ which arise therein, and use the conjecture to re-compute the Picard Lattice for the 95 families. Chapter 3 details the 95 calculations I made when computing the Picard lattices. In Chapter 4, I prove the existence of an index d embedding of Pic(S) into PicJ(S) for S with multisection index d. I include several appendices which contain the Mathematica code for my programs and various tables of quadratic forms and their values.