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I am familiar with (most of the) contents of Angelo Vistoli's notes on Descent theory (Stacks). I am also comfortable with basics of Schemes, their Cohomology (Cech), from Hartshorne's Algebraic geometry.

Now, I am interested to learn some basics on Moduli space/ Moduli problem/ Moduli stacks.

What is the logically next step towards this after knowing about stacks (not algebraic) and Scheme theory? What other prerequisites does one need to appreciate the moduli set up?

Any references are welcome.

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  • $\begingroup$ I have no interest/knowledge of Elliptic curves/Arithmetic geometry as of now. I might study something in future, but as of now, I know nothing about it.. $\endgroup$
    – Praphulla Koushik
    Aug 3, 2019 at 4:14
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    $\begingroup$ §4 of the Deligne-Mumford paper gives a self-contained introduction to moduli stacks, and you may browse through the rest of the paper for a compelling motivation. $\endgroup$
    – abx
    Aug 3, 2019 at 6:49
  • $\begingroup$ @abx Thanks. I looked at section 4... It looks nice.. I think I can understand 3/4 th of that section... I will ask if I have any specific question... $\endgroup$
    – Praphulla Koushik
    Aug 3, 2019 at 7:38
  • $\begingroup$ One example of how moduli spaces are 'used' is enumerative geometry. Enumerative problems are essentially intersection theory on moduli spaces, ranging from the more classical Schubert calculus (intersection theory on Grassmannians) to modern curve counting theories such as Gromov-Witten theory etc. $\endgroup$
    – loch
    Aug 6, 2019 at 9:54
  • $\begingroup$ @loch can you give some reference which you think is one of the well written introductions to that subject.. $\endgroup$
    – Praphulla Koushik
    Aug 6, 2019 at 11:14

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