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Über dieses Buch

This book focuses on a large class of geometric objects in moduli theory and provides explicit computations to investigate their families. Concrete examples are developed that take advantage of the intricate interplay between Algebraic Geometry and Combinatorics. Compactifications of moduli spaces play a crucial role in Number Theory, String Theory, and Quantum Field Theory – to mention just a few. In particular, the notion of compactification of moduli spaces has been crucial for solving various open problems and long-standing conjectures. Further, the book reports on compactification techniques for moduli spaces in a large class where computations are possible, namely that of weighted stable hyperplane arrangements (shas).

Inhaltsverzeichnis

Frontmatter

Chapter 1. Stable Pairs and Their Moduli

Abstract
The focus of these lectures is the higher-dimensional case, and it is hoped that the reader already has some familiarity with the one-dimensional case. So we will be rather brief.
Valery Alexeev

Chapter 2. Stable Toric Varieties

Abstract
For a detailed introduction to the theory of toric varieties, one should consult the usual sources [21, 48]. Our introduction is very brief and serves mainly to set up the notation and clarify the definitions (for example, our toric varieties are normal and do not have the “origin” fixed). The theory of stable toric varieties reviewed below is contained in [4, 9].
Valery Alexeev

Chapter 3. Matroids

Abstract
A matroid is a pair \( M = \left( {E,\,\mathcal{I}} \right) \) consisting of a (usually finite) set E and a set \( \mathcal{I} \subset 2^E \) of subsets called the independent sets. Equivalently, it can be defined using bases, or using the rank function \( r:\,2^E \to \mathbb{Z}_{\geq{0}}\).
Valery Alexeev

Chapter 4. Matroid Polytopes and Tilings

Abstract
Some of the results we explain here are contained in [22, 24]. Another good source on matroid polytopes is [53].
Valery Alexeev

Chapter 5. Weighted Stable Hyperplane Arrangements

Abstract
We give a brief introduction to Geometric Invariant Theory (GIT, for short) and to variations of GIT quotients (VGIT, for short). In general, GIT is a big and nontrivial theory, for arbitrary reductive groups G. However, the main point of this introduction is that, when G is a torus, the GIT quotients are very simple, and computing them is an easy combinatorial exercise.
Valery Alexeev

Chapter 6. Abelian Galois Covers

Abstract
Let \( \pi :\,X \to Y \) be a cyclic Galois cover of two varieties for the group \( G = \mu_n \) of roots of unity. Thus, we have a group action \(\mu_n \curvearrowright X \) and the quotient is Y. Let us assume that X and Y are smooth for now, until we understand how to deal with the general case.
Valery Alexeev

Backmatter

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