Cubic fourfolds with ADE singularities and K3 surfaces

Abstract

In this thesis, we will give a partial classification of cubic fourfolds by their isolated ADE singularities. We have a correspondence between cubic fourfolds and complete (2,3)- intersections in ℙ^4 having both certain isolated ADE singularities. The minimal model for a complete (2,3)-intersection in ℙ^4 with isolated ADE singularities is a quasi-polarized K3 surface of degree 6. We will prove that the existence of certain lattice embeddings into the K3 lattice is a necessary and sufficient condition for the existence of these singular cubic fourfolds and complete (2,3)-intersections, respectively. We will determine all direct sums of negative definite irreducible ADE lattices such that their direct sum with the rank one lattice whose generator has self-intersection number 6 admits a primitive embedding into the K3 lattice. This will prove the existence of complete (2,3)-intersections in ℙ^4 lying on smooth quadrics and having exactly these ADE singularities and their corresponding cubic fourfolds. Finally, we will show that we have an isomorphism between the moduli space of cubic fourfolds with certain ADE singularities and the moduli space of quasi-polarized K3 surfaces of degree 6 such that the quasi-polarization induces a birational map from the K3 surface into ℙ^4 whose image is a complete (2,3)-intersection in ℙ^4 having certain ADE singularities.