Soft question: beginners reference to moduli spaces
Here are some fairly recent and general references I like:
- The Handbook of Moduli http://intlpress.com/site/pub/pages/books/items/00000399/index.html
The article on logarithmic geometry by Abramovich et al. is great!
- The Isaac Newton Institute had a wonderful school on moduli spaces. You can find the videos here:
http://www.newton.ac.uk/event/mosw01
I highly recommend the series of lectures by Nitin Nitsure on deformation theory and Artin's criteria.
- The book based on the lectures given at that school just came out:
http://www.cambridge.org/US/academic/subjects/mathematics/geometry-and-topology/moduli-spaces
It has a wonderful article on stacks by Kai Behrend, working out the enlightening "moduli space of triangles" example in detail (with colorful pictures in the print version!).
I would advise:
- R. Hartshorne, "Deformation Theory", Springer.
Chapter 4 (in particular section 23) for an introduction to moduli problems, and introducing a little bit stacks as well. Here the focus is especially on moduli of curve.
- S. Kovacs, "Young person’s guide to moduli of higher dimensional varieties", https://sites.math.washington.edu/~kovacs/2013/papers/Kovacs__YPG_to_moduli.pdf
for an introduction to moduli problems in general and a detailed treatment of moduli of higher dimensional algebraic varieties.
The introductions of each chapter of
Geometry of Algebraic Curves, Volume II, Arbarello Enrico, Cornalba Maurizio, Griffiths Phillip
(the book itself is tuff, but the introductions are extremely intuitive and clear)