On Fano and Calabi-Yau varieties with hypersurface Cox rings

Abstract

This thesis contributes to the explicit classification of Fano and Calabi-Yau varieties.

First, we deal with complete intersections in projective toric varieties that arise from a non-degenerate system of Laurent polynomials. Here we obtain Bertini type statements on canonical and terminal singularities. This enables us to classify all non-toric terminal Fano threefolds that arise as a general complete intersection in a fake weighted projective space.

The second chapter is devoted to the classification of all smooth Fano fourfolds of Picard number two that have a general hypersurface Cox ring. Using the Cox ring based description of these varieties we investigate their birational geometry and compute Hodge numbers. Moreover, we present a toolbox for constructing examples of general hypersurface Cox rings including several factoriality criteria for graded hypersurface rings.

Finally, we give classification results on smooth Calabi-Yau threefolds of Picard number one and two that have a general hypersurface Cox ring.