UC Riverside

The Cox ring of an algebraic variety encodes important information on the birational geometry of the variety. When its Cox ring is finitely generated, a variety admits particularly desirable properties in the context of the Minimal Model Program and is called a Mori dream space. For example, all toric varieties are known to be Mori dream spaces so a natural next step in the problem is to study the birational geometry of projective varieties that can be constructed as blowups of toric varieties by studying their pseudoeffective cones and Cox rings. In this dissertation, we present a concrete criterion for the finite generation of the Cox ring of toric projective surfaces of Picard number one blown up at a smooth point using the coordinates of a polytope of the toric variety. We also present a criterion for the irreducibility of an effective divisor of the moduli space of n-pointed stable rational curves.