What is a moduli?

Question

I am aware that Question is broad. I come across names moduli spaces and vector bundles when reading introductions of books related to algebraic geometry. I am learning about vector bundles and want to know what is this moduli spaces are about.

I tried reading https://www.ma.utexas.edu/users/benzvi/math/pcm0178.pdf but it was not interesting so could not proceed. I am trying to read Theory of moduli by C.S. Seshadri in Proceedings of symposia in pure mathematics Algebraic geometry- Arcata 1974.

Any other introductory reference or some expository notes would be helpful in understanding this better. Some history and motivation would also be useful.

Answer 1

I'm just a student but here is some suggestions for reference :

• A first course in algebraic geometry, Harris.

This is a introductory book to algebraic geometry, so it's pretty basic but it contains a lot of various examples which are useful to keep in mind. I particularly advice reading chapter 4 (Families and parameter spaces) and chapter 21 (Parameter spaces and Moduli spaces).

• Introduction to Invariant and Moduli, Mukai.

This book assume basic knowledge of algebraic geometry, and is nicely introducing all the machinery necessary for studying geometric invariant theory and moduli, with always lot of examples and concrete computations.

• The Moduli Space of Curves and Its Tautological Ring, Ravi Vakil.

A very nice survey article about Moduli space of curves.