User david dynerman - MathOverflow

What elementary problems can you solve with schemes?

2011-03-21T15:48:21Z

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-contained problems that scheme theory answers - ideally something that I could explain to a fellow grad student in another field when they ask "What can you do with schemes?"

Let me give an example of what I'm looking for: In finite group theory, a well known theorem of Burnside's is that a group of order $p^a q^b$ is solvable. It turns out an easy way to prove this theorem is by using fairly basic character theory (a later proof using only 'elementary' group theory is now known, but is much more intricate). Then, if another graduate student asks me "What can you do with character theory?", I can give them this example, even if they don't know what a character is.

Moreover, the statement of Burnside's theorem doesn't depend on character theory, and so this is also an example of character theory proving something external (e.g. character theory isn't just proving theorems about character theory).

I'm very interested in learning about similar examples from scheme theory.

What are some elementary problems (ideally not depending on schemes) that have nice proofs using schemes?

Please note that I'm not asking for large-scale justification of scheme theoretic algebraic geometry (e.g. studying the Weil conjectures, etc). The goal is to be able to give some concrete notion of what you can do with schemes to, say, a beginning graduate student or someone not studying algebraic geometry.

Answer by David Dynerman for What can be said about a group from its presentation?

2010-08-16T15:02:57Z

Continuing the idea that working with bare presentations is "hard", automatic groups sometimes can give you a handle, if you're trying to study a specific group . Automatic groups are groups with finite state machines that can, essentially, solve the word problem for that group.

If you're studying a particular group and are lucky, a procedure such as Knuth-Bendix can compute an automatic structure for you. Then lots of hard computations become easy (e.g. the order of the group).

Magma has some of these algorithms implemented, see this Magma documentation page.

Answer by David Dynerman for What is the etymology for the term conductor?

2010-03-25T03:31:23Z

The first time I ever saw a conductor defined is not in the sense mentioned above, but in linear algebra, from Hoffman & Kunze's book. In their chapter on elementary canonical forms they define the conductor of vector $\alpha$ into a subspace $W$ with respect to a linear operator $T$ to be the ideal

$S_T(\alpha;W) = \{ g \in F[x] \mid g(T)\alpha \in W \}$

Where the ambient vector space is over the field $F$. Interestingly, they say that they themselves call this the 'stuffer' ideal (from German, das eistopfende Ideal), but claim that "Conductor" is more commonly used, and add that this term is

"preferred by those who envision a less aggressive operator $g(T)$, gently leading the vector $\alpha$ into $W$."

Hoffman & Kunze, Linear Algebra 2nd Edition, p. 201

Comment by David Dynerman

2013-04-06T18:24:31Z

It's helpful to remember that a graded ring S gives not only a projective scheme but also an embedding (via S(1)), and lots of rings can give the same projective scheme. If S is the ring you cite from Hartshorne, then S(1) gives the Segre embedding of your projective variety. I'm also interested in knowing if there are other easy rings to write down whose Proj gives you X x X. In the case when S is the polynomial ring k[x,y], then the total tensor product just gives you k[x,y,z,w], so you just get P^3 which isn't what we want.

Comment by David Dynerman

2011-03-28T19:29:44Z

The introduction to section 12.A in Isaacs Algebra: A Graduate Course talks a little about this view.