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I recently started studying algebraic geometry together with a couple of friends and especially in discussions online we keep reading about developing geometric intuition. There are some questions on this website about developing geometric intuition, but none of them really ask for what geometric intuition is. It doesn't seem obvious what the whole concept of geometric intuition means in the context of modern algebraic geometry, so it seems hard to judge when one has started developing it. Especially, when problems in number theory can often be related to algebraic geometry and then solve by using this geometric intuition.

Let me give an example from elementary analysis, where this is completely obvious. Take the squeeze theorem. Anyone can visualize two graphs in their head and "see" that anything between them must get pushed to the same point. Then the proof just corresponds to having learned how to translate a picture to a formal epsilon-delta argument. My question is then that is an algebraic geometer or arithmetic algebraic geometer working actually seeing nice pictures of lines, surfaces and curves in their head? Or is it just a matter of having developed experience with how different algebraic objects behave? The latter wouldn't seem any different from having developed intuition about, say, field theory through experience and this intuition could hardly be called "geometric" by anyone.

Feel free to close if this question is considered inappropriate for this website, I certainly understand. The reason for posting here instead of math.stackexchange is that graduate students in my department don't seem to have the experience themselves to answer it. I'm sure that this intuition keeps developing for a long time after finishing graduate studies. Hence, I was hoping for an experienced audience hopefully willing to answer.

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  • $\begingroup$ @Tim, traditionally questions like this which do not have a well-defined answer are made community wiki by checking the community wiki box. $\endgroup$ – Benjamin Steinberg Jan 19 '12 at 1:22
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    $\begingroup$ "My question is then that is an algebraic geometer or arithmetic algebraic geometer working actually seeing nice pictures of lines, surfaces and curves in their head?" The short answer is a plain "Yes" without any mental reservation. $\endgroup$ – Joël Jan 19 '12 at 1:51
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    $\begingroup$ Speaking as one who has worked at the far algebraic end of algebraic geometry (i.e. the part of algebraic geometry that is really commutative ring theory), I second Joel's unequivocal "yes". For those who work on more geometric problems, I'd expect the answer to be even more unequivocal, if "more unequivocal" were possible. $\endgroup$ – Steven Landsburg Jan 19 '12 at 6:33
  • $\begingroup$ @Joel: This is an answer, there is no reason to post this as a comment. $\endgroup$ – Martin Brandenburg Jan 19 '12 at 11:37
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    $\begingroup$ Dear Tim, Joel is correct; when people talk of developing geometric intuition, they mean developing a feeling for how curves, surfaces, etc. behave, and using this to analyze situations of interest. Some typical specific topics in which it is common to begin with little intuition, but to progressively develop more, are: blowing up, intersection theory, projective embeddings via very ample line bundles, semi-stable reduction, minimal models, divisors moving in linear systems, deformation theory, normalization, and etale morphisms. (Of course I could list many more.) Regards, Matthew $\endgroup$ – Emerton Jan 20 '12 at 8:38
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Vote this answer up if you consider yourself an algebraic geometer, and (in the course of your work) actually see nice pictures of lines, surfaces, and curves in your head.

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    $\begingroup$ Since the OP asks specifically about schemes and number theory, let me add that we also see nice pictures of nonreduced things (as thickenings of reduced things), of families (whose base may be "discrete", as Spec Z), maps (coverings come to mind) and probably more. $\endgroup$ – quim Jan 20 '12 at 10:10
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    $\begingroup$ This is certainly my experience. I suspect that if you did an MRI of working algebraic geometer, you'd see a lot of activity in the visual cortex. (Not that I'm volunteering.) $\endgroup$ – Donu Arapura Jan 20 '12 at 16:31
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Vote this answer up if you consider yourself an algebraic geometer, and (in the course of your work) do not see nice pictures of lines, surfaces, and curves in your head.

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    $\begingroup$ People voting this answer should leave their names, so we can later study them in detail... $\endgroup$ – Mariano Suárez-Álvarez Jan 20 '12 at 8:49
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    $\begingroup$ From The Unreal Life of Oscar Zariski (pg 109): "In support of his view of Zariski as a geometer for whom modern algebra could provide, at best, only a "temporary accommodation," Weil recalled a well-known anecdote about a conversation at a party between Zariski and Claude Chevalley [...]. The two men were discussing algebra versus geometry in algebraic geometry when at long last, exasperated, Zariski exclaimed, "But when someone says 'an algebraic curve,' surely you see something!" "Yes, of course," Chevalley quietly replied. "I see this: $f(x,y)=0$." $\endgroup$ – Samantha Y Nov 9 '15 at 16:11

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