PICARD LATTICES OF FAMILIES OF K3 SURFACES

ABSTRACT

Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.

ACKNOWLEDGMENTS

The author is grateful to I. Dolgachev for supervising this work, which was part of her Ph.D. dissertation at the University of Michigan. She would also like to thank W. Cherry for general consultations, J.H. Keum for pointing out an omission in a previous version, and R. Miranda and D. Cox for providing assistance via email.

Notes

While Batyrev's mirror construction does not always give the correct K3 mirror (see [7]), the spirit of his work suggests that Kreuzer/Skarke's 4319 toric K3 hypersurfaces [13] will mirror each other; the author is currently working to confirm this.)