I am a student almost without background on algebraic geometry (but I do know basic graduate algebra and topology). Now I am trying to understand something about algebraic stacks.
I want to start with this short AMS article first: http://www.ams.org/notices/200304/what-is.pdf
This article tries to define algebraic stacks by introducing the notion of the moduli
space of elliptic curves. However, I do not know anything about elliptic curves and
there are just too many possible books.
Can someone give some suggestions about how to proceed to learn enough elliptic curves so as to understand something about algebraic stacks? I also welcome other suggestions. Thank you.
Best Answer
You should really learn algebraic geometry first. After you're done with that, try reading Mumford's paper "Picard Groups of Moduli Problems", which is a fascinating window into the mind of one of the people who later invented algebraic stacks as he was himself figuring out what you are also trying to figure out. From that, you should have a sketch of how the properties of elliptic curves contribute to the properties of their moduli stack; make sure you work out or look up the various things he says about them. Then you can read a regular treatise on stacks.