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Results tagged with ag.algebraic-geometry
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user 290
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
108
votes
8
answers
15k
views
What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The Ati …
50
votes
Accepted
Given a polynomial f, can there be more than one constant c such that every root of f(x)-c i...
This is impossible by the Mason-Stothers theorem (which holds over any algebraically closed field of characteristic zero).
We want to find $f, g, h$ such that $f + g = h$ where $g$ is a constant and …
47
votes
Accepted
Grothendieck says: points are not mere points, but carry Galois group actions
Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) c …
44
votes
2
answers
3k
views
Why can't we take three loops?
Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names:
No l …
38
votes
6
answers
4k
views
Are all polynomial inequalities deducible from the trivial inequality?
I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of polynomial …
37
votes
Does module Hom commute with tensor product in the second variable?
You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only commu …
36
votes
Accepted
Why are polynomials so useful in mathematics?
Polynomials are, essentially by definition, precisely the operations one can write down starting from addition and multiplication. More formally, polynomials with coefficients in a commutative ring $R …
34
votes
2
answers
7k
views
What is the geometric meaning of integral closure?
More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its inte …
32
votes
4
answers
4k
views
Modular curves of genus zero and normal forms for elliptic curves
This is maybe the first question I actually need to know the answer to!
Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N …
31
votes
1
answer
4k
views
For which varieties is the natural map from the Chow ring to integral cohomology an isomorph...
My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to th …
29
votes
There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic for...
This is an extended comment on KConrad's discussion of symmetry groups. We can think of $k$-forms on a vector space $V$ (homogeneous polynomials of degree $k$) abstractly as elements of the symmetric …
28
votes
What is a field [Körper] really?
Fields are the simple (no nontrivial quotients) commutative rings. Grothendieck told us to work in nice categories with nasty objects rather than nasty categories with nice objects; fields are the nic …
28
votes
What is the field with one element?
Here's probably the simplest manifestation of the field-with-one-element phenomenon. Define a projective $n$-space of order $q$ to be a collection of points, lines, planes, etc. satisfying the usual …
26
votes
Why higher category theory?
Re: David Corfield's comment, this answer will mostly address "higher as in $(\infty, 1)$-categories" rather than "higher as in $n$-categories."
I also do understand you need the notion of abelia …
24
votes
Interesting results in algebraic geometry accessible to 3rd year undergraduates
This isn't a result so much as a perspective, but it is one of the main reasons I first got interested in algebraic geometry.
In basic algebraic number theory you learn that some extensions of the in …