Author: Eiko

Tags: algebraic geometry, moduli space

Time: 2024-10-21 18:24:26 - 2024-10-21 18:24:26 (UTC)

Overview

The set of certain objects we want to study or classify can be considered as a whole and form a ‘map’. Moduli spaces are magical maps that acquire extra non-trivial geometric structures (usually in an algebraic geometric sense) and are used to study the objects in a more geometric and systematic way.

Let’s say we have

• a category $\mathcal{C}$ consisting of objects we are interested in, and

• a notion of what is a family of objects $\pi :\mathcal{X}\to B$ of $\mathcal{C}$ over any base space $B$.

Universal Family

We might try to find a family $\pi :\mathcal{X}\to B$

• whose fibres contains all possible objects (up to isomorphism) in $\mathcal{C}$

• a notion of equivalence in $B$ such that $b_1\sim b_2$ if and only if the fibres ${\mathcal{X}}_{b_1}\cong {\mathcal{X}}_{b_2}$.

Then $\mathcal{M}:=B/\sim$ is the moduli space of objects in $\mathcal{C}$ we are looking for, and $

The best case is that we have some invariants that partition the moduli space into finite type spaces

$$\mathcal{M}=\underset{i\in I}{⨆}{\mathcal{M}}_i$$

for example the space of smooth projective curves has a discrete invariant genus, and the moduli space of curves is

$$\mathcal{M}=\underset{g\ge 0}{⨆}{\mathcal{M}}_g$$

where ${\mathcal{M}}_g$ is the moduli space of curves of genus $g$, each of which has $3g-3$ parameters.

Three Aspects of Moduli Spaces

• Existence and construction of moduli spaces

  This include things like stacks, GIT, Hilbert schemes, fine moduli spaces, coarse moduli spaces, etc.

• Compactification of moduli spaces

  Semi-stable reduction, wall-crossing, toroidal compactification, etc.

• Applications of moduli spaces

  Enumerative geometry, invariant computation, mirror symmetry, representation theory, combinatorics, arithmetic statistics, etc.