UC Berkeley

In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety $X$ over $\mathbb{C}$ and the algebraic, geometric, and cohomological properties of divisors on $X$. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program.

In chapter 2 we explore criteria for Totaro's notion of $q$-amplitude. A line bundle $L$ on $X$ is $q$-ample if for every coherent sheaf $\mathcal{F}$ on $X$, there exists an integer $m_0$ such that $m\geq m_0$ implies $H^i(X,\mathcal{F}\otimes \mathcal{O}(mL))=0$ for $i>q$. We show that a line bundle $L$ on a complex projective scheme $X$ is $q$-ample if and only if the restriction of $L$ to its augmented base locus is $q$-ample. In particular, when $X$ is a variety and $L$ is big but fails to be $q$-ample, then there exists a codimension $1$ subscheme $D$ of $X$ such that the restriction of $L$ to $D$ is not $q$-ample.

In chapter 3 we study the singularities of Cox rings. Let $(X,\Delta)$ be a log Fano pair, with Cox ring $R$. It is a theorem of Birkar, Cascini, Hacon and McKernan that $R$ is finitely generated as a $\C$ algebra. We show that Spec $R$ has log terminal singularities.