I have been studying mathematics for 4 years and I know schemes (I studied chapters II, III and IV of Hartshorne). I would like to learn some moduli theory, especially moduli of curves.
I began reading "Harris, Morrison – Moduli of curves", but I found it very vague. Although I like their geometric ideas, I would like to have also a rigorous reference for the results stated by them. May someone suggest to me a book, lecture notes, or an expository paper where I can find a precise introduction to moduli theory?
Thanks!
Best Answer
Manetti's online notes are among the best introductions to moduli theory: they contain some very explicit and down-to-earth calculations, like the determination of the deformations of the Segre-Hirzebruch surfaces.
Since you have been studying Hartshorne's Algebraic Geometry book, you might like his notes on deformation theory, which are more algebraic than Manetti's (they have become a Springer book , but the free online notes are not very different: exercises have been added to the book and that's about it).
If you are ambitious you could try (maybe a little later) the very comprehensive treatise by a respected specialist : Sernesi's Deformations of algebraic schemes , volume 334 in Springer's prestigious series Grundlehren der Mathematischen Wissenschaften .
If you are interested in moduli of vector bundles on algebraic varieties, here is an online introduction by Miró-Roig.
And if you want to learn, more generally, about moduli spaces of coherent sheaves, just download the book by Huybrechts-Lehn :
www.math.uni-bonn.de/people/huybrech/moduli.ps
Good luck!