Rational Curves on Algebraic Spaces and Projectivity Criteria

Abstract

In Chapter 1, we provide background material that is relevant to the remainder of the thesis. The aim is to make the thesis accessible to readers whose background in algebraic geometry is at the level of one of the standard introductory graduate textbooks. The topics summarised here include group actions on schemes and the quotients of such actions, positivity conditions of divisors, the Kleiman-Mori cone, and the Minimal Model Program. We also give a brief and informal introduction to algebraic spaces; while not completely rigorous, it should be enough to understand the role of algebraic spaces in Chapter 3. \\\indent In Chapter 2, we give a short proof of a Grothendieck-Lefschetz Theorem for equivariant Picard groups of nonsingular varieties with the action of an affine algebraic group. \\ \indent Chapter 3 is the core of the thesis. We show that a smooth Moishezon space $Y$ is non-projective if and only if it contains a rational curve such that $-[C] \in \overline{\mathrm{NE}}(Y)$. More generally, this holds if $Y$ has $\mathbb{Q}$-factorial, log terminal singularities. We derive this as a consequence of our main technical result: that we can run the relative minimal model program when the base is an algebraic space $Y$ of finite type over a field of characteristic $0$. As a second application, we show that every log canonical pair $(Y, \Delta)$, where $Y$ is an algebraic space of finite type over a field of characteristic $0$ admits a dlt modification that is projective over $Y$.