Cox Rings and Partial Amplitude

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