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.
2026-03-28 22:13:44.1774736024
On
On
On
On
Visualizations of some of the abstractions of algebraic geometry
1.9k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
6
There are 6 best solutions below
0
On
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.
4
On
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.
0
On
The books "Geometry of Schemes" and "The red book of Varieties and Schemes" are great to get intuition about this.
0
On
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.
Mumford's The Red Book of Varieties and Schemes is quite famous for its pictures and the emphasis on geometric intuition.