Moduli Spaces
One of the most fascinating features of geometry is that collections of particular geometric objects often come with a natural geometric structure themselves. For example, consider the collection of conics in P^2. A conic is determined by its defining equation, which is a degree 2 homogeneous polynomial in 3 variables, and hence is determined by its 6 coefficients. Since these coefficients cannot be simultaneously vanishing and are only well-defined up to an overall scaling, we see that the set of conics in P^2 is in natural bijection with the projective space P^5. Hence, it has a natural structure as an algebraic variety (not just a set).
This is a simple example of what is called a moduli space. A moduli space is a space – topological, differential, algebraic, etc. – each point of which corresponds to a certain geometric object, and whose additional geometric structure in some way reflects how these objects relate to each other. This is made precise via the functorial point of view, originally introduced in algebraic geometry by Grothendieck and his collaborators. This provides a rigorous definition for the terms “moduli problem” and “moduli space”.
In this talk I will provide a gentle introduction to moduli spaces, focusing mostly on the moduli space of curves. I will introduce the functorial point of view and discuss representability, as well as explaining why automorphisms prevent the existence of a fine moduli space.