Intersection theory on Spherical Varieties

The aim of this thesis is to do intersection theory on spherical varieties, i.e. varieties on which a reductive group $G$ acts with an open $B$-orbit, where $B$ is a Borel subgroup of $G$. It turns out that on these varieties intersection theory is easier (for a smooth complete spherical variety over $\mathbb{C}$ it is its simplicial cohomology), and in fact the action of the group gives us an explicit description of how the Chow ring $A^*(X)$ works. In the first chapter we will recall what we need for the rest of the thesis about intersection theory: the definition of the intersection product and how the Chow rings are related to the cohomology, if the field is $\mathbb{C}$. The second one is really the hearth of the thesis: we give a set of generators for the Chow groups of a spherical variety and we describe how intersection product works. In the third chapter we study the varieties of the form $G/P$, where $G$ is a semisimple group and $P$ a parabolic subgroup of it. In this setting the Bruhat decomposition allows us to calculate the Chow group, and we will find another description of the Chow ring of $G/P$. In the fourth chapter we discuss the geometry of spherical varieties: the main results will be that those varieties can be described combinatorially. In fact, we can associate to a spherical variety a fan, which can be used to describe our variety both locally and globally: we can translate in a combinatorial language questions such as when this variety is quasiprojective? When it is proper? Moreover, we will describe morphisms between spherical varieties using these fans. The main examples of spherical varieties are toric varieties, Grassmannians and complete symmetric varieties. The fifth chapter is mostly aimed at defining the Halphen ring of a homogeneous space in a more friendly environment (which is more explicit than the general one). But what is the Halphen ring: the idea is the following. Assume that we want to do intersection theory on a homogeneous spherical variety, for example $(\mathbb{C}^*)^n$. Then the Chow ring does not give us a lot of information because it is always 0 but in degree $n$, so we need something else: this somethig else is the Halphen ring. The idea is that if we consider all the compactifications $p:X\to (\mathbb{C}^*)^n$ where $X$ is a spherical variety, we can consider $\displaystyle{\lim_{\rightarrow} A^*(X)}$ instead of simply $A^*((\mathbb{C}^*)^n)$. This ring is much bigger than $A^*((\mathbb{C}^*)^n)$; it will describe better how ``intersection theory'' works, and we will give a description of this ring using the action of $G$. Actually this intersection theory depends on the action of the group on our spherical variety, see the examples of this section. In the last chapter we will talk about intersection theory on another particular case of spherical varieties: toric varieties. We will find ways to compute the cap product with combinatorial methods and we will give a combinatorial description of both the Chow ring of a complete smooth toric variety, and the Halphen ring of $(\mathbb{G}_m)^n$

Moduli of Q\mathbb{Q}-Gorenstein pairs and applications

We develop a framework to construct moduli spaces of $\mathbb{Q}$-Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of $\mathbb{Q}$-stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than $\frac{1}{2}$. Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.Comment: Improved exposition and minor corrections throughout. Final version to appear in Journal of Algebraic Geometr

Degenerations of twisted maps to algebraic stacks

We give a definition of twisted map to an algebraic stack with projective good moduli space, and we show that the resulting functor satisfies the existence part of the valuative criterion for properness.Comment: 23 pages, comments very welcome

Wall crossing for moduli of stable log pairs

We prove, under suitable conditions, that there exist wall-crossing and reduction morphisms for moduli spaces of stable log pairs in all dimensions as one varies the coefficients of the divisor.Comment: Improved exposition and minor corrections throughout. Final version to appear in Annals of Mathematic

A criterion for smooth weighted blow-downs

We establish a criterion for determining when a smooth Deligne-Mumford stack is a weighted blow-up. More precisely, given a smooth Deligne-Mumford stack $\mathcal{X}$ and a Cartier divisor $\mathcal{E} \subset \mathcal{X}$ such that (1) $\mathcal{E}$ is a weighted projective bundle over a smooth Deligne-Mumford stack $\mathcal{Y}$ and (2) for every $y\in\mathcal{Y}$ we have $\mathcal{O}_{\mathcal{X}}(\mathcal{E})|_{\mathcal{E}_y}\simeq \mathcal{O}_{\mathcal{E}_y}(-1)$, then there exists a contraction $\mathcal{X}\to\mathcal{Z}$ to a smooth Deligne-Mumford stack $\mathcal{Z}$. Moreover, $\mathcal{X}$ can be recovered as a weighted blow-up along $\mathcal{Y}\subset \mathcal{Z}$ with exceptional divisor $\mathcal{E}$. As an application, we show that the moduli stack $\overline{\mathscr{M}}_{1,n}$ of stable $n$-pointed genus one curves is a weighted blow-up of the stack of pseudo-stable curves. Along the way we also prove a reconstruction result for smooth Deligne-Mumford stacks that is of independent interest.Comment: 27 pages, comments welcome