In-Person and Online Talk 

We are interested in characterising when Y is projective. Peternell conjectured that if Y contains no rational curves, then Y is  projective. Peternell's conjecture was proved by Shokurov and Birkar-Cascini-Hacon-McKernan using MMP techniques. This, however, does not characterise projectivity; indeed, there exist plenty of projective varieties with plenty of rational curves. In this talk, we will explain a characterisation of projectivity in terms of the presence of badly-behaved rational curves. The proof of this projectivity criterion hinges on being able to run an MMP in which the base is an algebraic space, rather than a quasi-projective variety.