Picard Lattices of Families of K3 Surfaces
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چکیده
Picard Lattices of Families of K3 Surfaces bysarah-marie belcastro Chair: Igor Dolgachev It is a nontrivial problem to determine the Picard Lattice of a given surface; theobject of this thesis is to compute the Picard Lattices of M. Reid’s list of 95 fami-lies of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projectivespace. Reid’s list arises in many problems; here we look at an application to MirrorSymmetry.One can define an analogue of Mirror Symmetry for Calabi-Yau threefolds forK3 surfaces. This analogue coincides with the “strange duality” for the 14 surfacesingularities of V. I. Arnold. Here we investigate the Mirror Symmetry of Reid’s 95families of surfaces; Arnold’s singularities are on this list.Denote a surface by S and a mirror family by Š. Then we define a mirror familybyPic(S)H2(S,Z) = Pic(Š) ⊥ U.By computing the Picard Lattice for each of these 95 surfaces, I am able to determinewhether the mirror family for each one is also on Reid’s list.
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