All Questions
22,546 questions
31
votes
7
answers
10k
views
Quotients of Schemes by Free Group Actions
I've often seen people in seminars justify the existence of a quotient of a scheme by an algebraic group by remarking that the group action is free. However, I'm pretty sure they are also invoking ...
- 5,548
10
votes
3
answers
2k
views
Unstable Vector Bundles
As a follow up to me other question, what can be said about unstable vector bundles? I know this is rather open ended, but what sorts of horrible things does having a subbundle of strictly greater ...
- 16k
4
votes
2
answers
2k
views
Pushforwards of Line Bundles and Stability
I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a ...
- 16k
8
votes
1
answer
688
views
Universal covers of domains in complex projective space
The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
- 4,485
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 ...
- 118k
6
votes
2
answers
1k
views
Does projectiveness descend along field extensions?
Background: Properness is a much more robust notion than projectiveness. For example, properness descends along arbitrary fpqc covers (see, for example, Vistoli's Notes on Grothendieck topologies, ...
- 24.1k
11
votes
4
answers
2k
views
Moduli spaces of complex curves as algebraic varieties
Using a minimum of technical vocabulary, give a summary of why it is that the moduli space of genus g complex curves with n marked points has a natural compactification that is isomorphic (as a ...
- 7,062
17
votes
2
answers
3k
views
Are curves with `fractional points' uniquely determined by their residual gerbes?
One makes precise the vague notion of "curve with a fractional point removed" (see for instance these slides) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are ...
- 10.5k
13
votes
5
answers
5k
views
Examples and intuition for arithmetic schemes
How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...
6
votes
1
answer
777
views
Existence of proper regular models for varieties over Q and other global fields
What is known about regular proper models for smooth projective varieties over Q? Results for other global fields would also be interesting, as well as general comments and suggested references for ...
- 5,637
15
votes
7
answers
5k
views
Morphisms of (quasi-)projective varieties
This is another "homework help" question, which is still hopefully of at least pedagogical interest to working mathematicians.
So, I'm currently taking an intro algebraic geometry class, and one ...
- 12.6k
6
votes
3
answers
1k
views
Is there a software package that does Schubert Calculus computations?
Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...
- 16k
17
votes
3
answers
1k
views
R2 and S3 for rings.
For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition ...
- 273
20
votes
1
answer
2k
views
Functorial characterization of open subschemes?
Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...
- 5,407
13
votes
6
answers
3k
views
Gromov-Witten theory and compactifications of the moduli of curves
Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather ...
- 21k
129
votes
15
answers
51k
views
A learning roadmap for algebraic geometry
Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be ...
3
votes
1
answer
514
views
4
votes
1
answer
1k
views
The existence of primitive and sufficiently ample line bundles on K3 surfaces?
Let S be a surface and L be a line bundle on S. For any zero-dimensional closed subschemes x of S, there is natural map from global sections of L to the global sections of L restricting to x (which is ...
- 41
62
votes
8
answers
14k
views
Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
user709
6
votes
5
answers
3k
views
Is very ampleness of a divisor on a curve determined entirely by degree and genus?
Edit: Apparently the answer is "no", so what is an example of two curves of genus g, and a divisor of degree d on each, such that one is very ample and the other is not?
Question as originally stated:...
- 11.3k
22
votes
13
answers
8k
views
Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
13
votes
1
answer
1k
views
What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?
Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian ...
- 2,270
6
votes
2
answers
1k
views
Explicit Direct Summands in the Decomposition Theorem
Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...
- 8,866
20
votes
2
answers
4k
views
"Fermat's last theorem" and anabelian geometry?
Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...
- 10.8k
9
votes
1
answer
1k
views
Example where you *need* non-DVRs in the valuative criteria
The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it ...
- 24.1k
12
votes
4
answers
2k
views
For which hypersurfaces in projective space does the complement admit an algebraic group structure?
For example, if $H$ is a hyperplane, then $\mathbb{P}^n - H = \mathbb{A}^n$, which is a vector space.
If $n = m^2 - 1$, then we can regard $\mathbb{A}^{n+1}$ as the space of $m \times m$ matrices and ...
- 3,918
14
votes
5
answers
2k
views
Rational maps with all critical points fixed
What can be said about rational self-maps of $\mathbb P^1$
for which all critical points are also fixed points ?
If all but one of the fixed points are critical, there is
a characterization in http://...
- 8,828
5
votes
1
answer
586
views
a general theory of configurations?
Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction ...
- 10.8k
9
votes
3
answers
3k
views
Why is the Euler characteristic of powers of a line bundle a polynomial in the power?
Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic $\chi(L^k)$ of tensor ...
- 31.2k
6
votes
2
answers
1k
views
Higher vanishing cycles
The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-...
- 10.8k
27
votes
7
answers
6k
views
Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
- 52.7k
43
votes
1
answer
19k
views
What is inter-universal geometry?
I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
- 10.8k
6
votes
2
answers
468
views
Algorithms for semistable reduction of families of curves
This is a somewhat vague question which came up MSRI a few days ago: Suppose I have a family of curves over a one dimensional base, given in a computationally explicit way. For example, maybe I have a ...
- 156k
11
votes
3
answers
1k
views
Is there a version of the valuative criteria for separateness/properness for varieties?
What I had in mind was something like the following:
X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way.
Is there a good reason why ...
- 4,612
80
votes
7
answers
20k
views
Teaching statements for math jobs?
What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I am applying for jobs, and I need to write one of these ...
16
votes
2
answers
2k
views
Bad Categorical Quotients
Let $G$ be an algebraic group acting on a scheme $X$. Then $f: X \to Y$ is called a categorical quotient if it is constant on $G$-orbits and every $X \to Z$ constant on $G$-orbits factors through it ...
- 2,806
15
votes
7
answers
2k
views
Examples of rational families of abelian varieties.
I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...
- 10.5k
14
votes
4
answers
3k
views
References for syntomic cohomology
Could anyone point to good readable references for learning about syntomic cohomology?
- 5,637
6
votes
4
answers
1k
views
When is a map given by a word surjective?
Let $w(x,y)$ be a group word in $x$ and $y$.
Let $x$ and $y$ now vary in $\operatorname{SL}_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of ...
- 20.2k
38
votes
6
answers
6k
views
"Points" in algebraic geometry: Why shift from m-Spec to Spec?
Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started ...
- 2,967
25
votes
4
answers
2k
views
algebraic group G vs. algebraic stack BG
I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
- 24.1k
22
votes
6
answers
3k
views
Formal consequences of Riemann-Roch (multiple answers welcome)
This question aims to pin down what Riemann-Roch can tell us about a divisor on a curve, without any "geometric thinking". It can be annoying to wonder if there is some clever trick you're missing ...
- 11.3k
8
votes
1
answer
2k
views
Mirror symmetry for noncompact Calabi-Yau manifolds
In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our ...
- 287
11
votes
2
answers
2k
views
Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 5,637
11
votes
8
answers
3k
views
Are good introductory/pedagogical problems in algebraic geometry rare?
I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a ...
16
votes
3
answers
5k
views
Stalks of sheaf-hom
Let $F$ and $G$ be sheaves on $X$. Under what conditions is the natural map from the stalk at $p$ of $\mathcal{H}\kern{-1pt}\mathit{om}(F,G)$ to $\mathrm{Hom}(F_p, G_p)$ an isomorphism?
- 161
14
votes
2
answers
882
views
A complex manifold which is quasiprojective in two different ways
Does there exist a complex manifold M which is a quasiprojective variety in two "essentially" different ways? Let me be more specific. I'm looking for a complex manifold M together with two ...
- 44.8k
16
votes
2
answers
2k
views
"synthetic" reasoning applied to algebraic geometry
A hyperlinked and more detailed version of this question is at
nLab:synthetic differential geometry applied to algebraic geometry.
Repliers are kindly encouraged to copy-and-paste relevant bits of ...
- 19.8k
10
votes
2
answers
944
views
Logarithmic structures on moduli of elliptic curves over Z
I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...
- 52.7k
30
votes
5
answers
4k
views
Deformation theory of representations of an algebraic group
For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
the obstruction to deforming V as a representation of G is an element of H2(G,V&...
- 24.1k