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What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces? (Possibly introducing stacks also)? I'm looking for something which really gets the pictures across and helps build my beginners intuition. However it should be something that is not a totally trivial read either (i mean i'd like to be able to actually use the results in that paper/ short book also).

If it helps: the direction I'm looking into the subject will primarily be from the point of view of analytic varieties and secondarily from the point of view of (schemes, or generalizations thereof (provided there is an initial introductory portion on that generalization)).

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Here are some fairly recent and general references I like:

  1. 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!

  1. 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.

  1. 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!).

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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.

for an introduction to moduli problems in general and a detailed treatment of moduli of higher dimensional algebraic varieties.

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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)

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    $\begingroup$ The book is a light porous rock? $\endgroup$ – user50229 Aug 26 '16 at 9:36
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For moduli I would second Geometry of Algebraic Curves, Volume II.

For a clear introduction to stacks, I like Martin Olsson's book "Compactifying Moduli Spaces for Abelian Varieties". A good motivation for studying stacks is Mumford's beautiful paper "Picard groups of moduli problems".

Also there is a book being written on stacks by several authors (Behrend, Conrad, Edidin, Fulton, Fantechi, Göttsche, Kresch) that is extremely clear: http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1

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    $\begingroup$ Unfortunately, I have heard that the book won't be completed. $\endgroup$ – Martin Brandenburg Oct 6 '14 at 21:33
  • $\begingroup$ Oh really? That's too bad, the notes so far are fantastic. $\endgroup$ – rfauffar Oct 7 '14 at 1:08
  • $\begingroup$ The book is indeed a brother of "Methods of Homological Algebra: Vol. II". Well, we still have drafts and, say, Behrend's lectures mentioned by Piotr. $\endgroup$ – Anton Fonarev Oct 8 '14 at 21:24

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