Free curves on varieties

In this thesis we study various ways in which every two general points on a variety can be connected by curves of a fixed genus, thus mimicking the notion of a rationally connected variety but for arbitrary genus. We assume the existence of a covering family of curves which dominates the product of a variety with itself either by allowing the curves in the family to vary in moduli, or by assuming the family is trivial for some fixed curve of genus g. A suitably free curve will be one with a large unobstructed deformation space, the images of whose deformations can join any number of points on a variety. We prove that, at least in characteristic zero, the existence of such a free curve of higher genus is equivalent to the variety being rationally connected. If one restricts to the case of genus one, similar results can be obtained even allowing the curves in the family to vary in moduli. In later chapters we study algebraic properties of such varieties and discuss attempts to prove the same rational connectedness result in positive characteristic.This thesis is not currently available via ORA