Where, or do there exist, good visualizations of sheaves, stalks, stacks, and/or schemes? I'm a better visual thinker than I am a symbolic thinker, and it would be easier for me to follow some of the algebraic geometry papers that have no pictures whatsoever if I had some kind of visual intuition of what these objects look like.
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$\begingroup$ By what means are you learning about these structures? Presumably you at least have a book. Doesn't that book have a single example? $\endgroup$– Ryan BudneyNov 8, 2010 at 5:16
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$\begingroup$ I do not have a book. I have been looking mostly at the arXiv. Googling for "Introduction to Sheaves" reveals papers and course notes that usually start from a category-theory point of view and don't have useful pictures. $\endgroup$– graveolensaNov 8, 2010 at 5:37
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4$\begingroup$ You're trying to learn algebraic geometry without a book?!? You might as well try to learn trapeze without a safety net. There's no reason to do that: get yourself one of the introductory texts mentioned in the answers below. $\endgroup$– Pete L. ClarkFeb 7, 2011 at 6:36
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6 Answers
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David Einsenbud and Joe Harris' book The Geometry of Schemes has a large number of really nice pictures which could help you gain a little bit of geometric intuition. That being said, it is also an excellent book which you should check out if you are interested in schemes.
answered Nov 8, 2010 at 13:36
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Mumford's The Red Book of Varieties and Schemes is quite famous for its pictures and the emphasis on geometric intuition.
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I've just started with this stuff, so take this for what it's worth.
What's been working for me is to imagine a sheaf as the sheaf of continuous functions from the reals to the reals. The restriction maps are just the normal restrictions of functions to open sets.
For presheaves I think of the presheaf of bounded continuous functions from reals to reals.
Stalks are just the germs thereof. I draw an element of a stalk by drawing coordinate axes, choosing a point x on the horizontal axis, and then draw the graph of a continuous function on a tiny neighborhood of x. The neighborhood is supposed to be "infinitesimal." The stalk is the set of all such graphs on that neighborhood.
answered Nov 8, 2010 at 23:41
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$\begingroup$ Another good example of a sheaf is the sheaf associated to a function. So you can view a sheaf as a degenerate analogue of a fibration. $\endgroup$ Nov 9, 2010 at 3:16
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$\begingroup$ @Ryan: Dear Ryan, Do you mean "the sheaf associated to a fibration"? $\endgroup$– Matt ENov 9, 2010 at 3:52
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$\begingroup$ @Matt E: I'm talking about what Bredon would call "The Leray Sheaf of a map of topological spaces". This is what Burt Totaro used in one of his papers to study the homology of configuration spaces in complex projective varieties. It's very nice in the special case of a fibration but still a reasonable book-keeping device for plain old maps. This is IV $\S 4$ in Bredon's "Sheaf theory" book. $\endgroup$ Nov 9, 2010 at 19:01
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$\begingroup$ @Ryan: Dear Ryan, Thanks, now I understand: "a function" here means any map of spaces. (I was thinking of a scalar valued function for some reason, and got confused.) $\endgroup$– Matt ENov 10, 2010 at 1:09
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Algebraic varieties are a lot like complex analytic spaces, which are special kinds of manifolds. So if you develop intuition about geometrical objects equipped with complex analytic functions, then it would help you a great deal in intuition on varieties.
Another way is to get intuition on curves, which are simpler than the rest. And smooth complex projective curves are like compact Riemann surfaces, which can be visualized.
You could again look at pictures such as that of $Spec\ \mathbb Z$ and $Spec\ \mathbb Z[x]$ given in Mumford's Red book.
The functor of points approach is also quite a nice way to think. Roughly, a scheme can be specified by its $R$-valued points for every ring $R$.
A concrete example of a stalk is the notion of germs of holomorphic functions, which might be familiar from complex analysis.
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The paper "What is a Sheaf?" is not bad at all, it even has a picture or two. It predates the arxiv though, but not Jstor. It is by Seebach, Seebach and Steen, I believe. If you can't find it let me know.
answered Jan 25, 2011 at 14:50
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$\begingroup$ The paper probably predates Jstor too, but was scanned and put into it nonetheless! :) $\endgroup$ Feb 7, 2011 at 6:37
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The books "Geometry of Schemes" and "The red book of Varieties and Schemes" are great to get intuition about this.