Construction of Non-Trivial Deformations via Frobenius

Overview

In the second lecture, we discuss a technique of S. Mori to construct deformation of morphisms via reduction modulo p, and show the existence of rational curves on smooth projective varieties whose canonical divisors are not nef.

This technique, developed in the famous solution [Mori 1] of a conjecture of R. Hartshorne, was the starting point to the theory of extremal rays and minimal models. As one of its applications, we characterize a class of varieties which contain sufficiently many rational curves (a criterion for uniruledness in terms of canonical divisors).

Because of the nature of Mori’s technique, we work in the category of schemes: everything in this section is algebraic.