Algebraic varieties are of great interest within pure mathematics and are connected with many problems in number theory, geometry and mathematical physics. Within this project we focus on the interaction of two particular classes of algebraic varieties, namely varieties with trivial first Chern class and algebraic curves. This is inspired by the well-known result of Lazarfeld that connects the Brill-Noether theory of a curve C and the geometry of a K3 surface containing it.
The minimal model program reduces the birational classification of smooth projective varieties to the study of three classes: Fano varieties, varieties with trivial first Chern class (Ricci flat varieties) and canonically polarised varieties. In this project we will also include singular varieties and study the respective counterparts of these classes.
Our aims are: to produce new examples of Ricci flat varieties; to make progress towards their classification in small dimension; to study the (moduli of) projective curves; to find pairs of singular plane curves which are naturally linked with surfaces isogeneous with a product of curves, having unexpected Betti numbers.