We develop some foundational results in a higher dimensional foliated Mori theory, and

show how these results can be used to prove a structure theorem for the Kleiman-Mori cone

of curves in terms of the numerical properties of $K_{\cal F}$ for rank 2 foliations

on threefolds. We also make progress

toward realizing a minimal model program for rank 2 foliations on threefolds.

The Kawamata-Morrison cone conjecture have attracted a lot of attention in algebraic geometry. In this thesis, the author will deal with this conjecture for Calabi-Yau 3-folds whose nef cone is polyhedral in chapter 3. In chapter 4, we will investigate the null cone and the automorphism group of Calabi-Yau 3-folds of Picard number $4$, which serves as a very good example for understanding the Kawamata-Morrison conjecture.

A log del Pezzo surface is a klt projective surface whose canonical divisor is anti-ample. We classify all log del Pezzo surfaces of Picard number one defined over algebraically closed fields of characteristic different from two and three. We also discuss some consequences of the classification. For example, we show that log del Pezzo surfaces defined over algebraically closed fields of characteristic higher than five have at most four singular points and admit a log resolution that lifts to characteristic zero over a smooth base.

In this dissertation, we study the problem of the deformation invariance of plurigenera of algebraic varieties using techniques from the Mori and Iitaka Programs. In characteristic zero, we study the problem for smooth families of non-negative Kodaira dimension. In this case, we reformulate a famous theorem of Siu in terms of a condition on the central fiber of certain special models of the relative Iitaka fibration of X over T. Under some additional assumptions, this gives algebraic proofs of deformation invariance of plurigenera, generalizing results of Nakayama and Kawamata. In positive and mixed characteristic, we construct examples of families of smooth surfaces where all sufficiently divisible plurigenera fail to be constant, answering a quesion of Katsura and Ueno. In particular, invariance of plurigenera does not follow from the MMP and Abundance Conjecture. Lastly, we show that invariance of all sufficiently divisible plurigenera holds for families of quasi-elliptic surfaces and certain families of log Calabi-Yau fibrations of small relative dimension.

We study the problem of whether repeated normalized Nash blowups resolve toric singularities. We first review results of Gonzalez-Sprinberg, describing the normalized Nash blowup combinatorially and proving resolution for toric surfaces. We then investigate the problem in higher dimensions, letting $X$ be an affine toric variety of dimension at least $3$ and considering each invariant divisor $Y$ of its normalized Nash blowup $\text{NNB}(X)$. For the case where $Y$ is the strict transform of an invariant divisor $Y_0$ of $X$, we show that the birational map from $Y$ to $\text{NNB}(Y_0)$ is a morphism. For the case where $X$ is $3$-dimensional and $Y$ is any invariant divisor of $\text{NNB}(X)$, we prove the following result. Let $\tau$ be the ray corresponding to $Y$ in the fan of $\text{NNB}(X)$, and let $Z$ be the surface corresponding to $\tau$ in the stellar subdivision (also known as the star subdivision) of the fan of $X$ along $\tau$. Let $\text{MR\textsuperscript{+}}(Z)$ be the surface obtained from the minimal resolution of $Z$ by blowing up all invariant points. Then the birational map from $\text{MR\textsuperscript{+}}(Z)$ to $Y$ is a morphism. For the case where $X$ is a $3$-dimensional quotient singularity and $Y$ is the strict transform of an invariant divisor $Y_0$ of $X$, we deduce that the singularities of $Y$ are milder than those of $Y_0$.

Formal schemes are (topologically) ringed spaces that Grothendieck introduced in EGA. They are simultaneously analogues for admissible rings of schemes for general commutative rings and a “bridge” between analytic and algebraic geometry. More recently, formal schemes have been a subject of interest in studies of rigid analytic geometry, where, due to work by Raynaud, Bosch, and Lütkebohmert, they act as models for rigid analytic varieties. The recent interest in these objects has led to study of them in their own right. In this thesis, we investigate whether a minimal model program could exist for formal schemes.

For C a smooth curve and ξ a line bundle on C, the moduli space UC(2, ξ) ofsemistable vector bundles of rank two and determinant ξ is a Fano variety. We show that UC(2, ξ) is K-stable for a general curve C ∈ Mg. As a consequence, there are irreducible components of the moduli space of K-stable Fano varieties that are birational to Mg. In particular these components are of general type for g ≥ 22.