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Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

30
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6answers
4k 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 ...
21
votes
4answers
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 ...
21
votes
6answers
2k 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 ...
7
votes
1answer
1k 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 ...
10
votes
2answers
1k 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, ...
7
votes
8answers
2k 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 ...
8
votes
3answers
2k 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 $SheafHom(F,G)$ to $Hom(F_p, G_p)$ an isomorphism?
11
votes
2answers
507 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 ...
15
votes
2answers
1k 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 ...
8
votes
2answers
784 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 ...
27
votes
5answers
3k 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&...
8
votes
2answers
554 views

What is the affinization of M_g?

This question is inspired by What is an example of a function on M_g? . Consider Mg, the moduli space of genus g curves, NOT compactified. When g is 3 or greater, this is not affine. Does anyone know ...
6
votes
1answer
648 views

Limit Linear Series

A linear series on a curve C is a line bundle L together with a subspace V of the global sections of L. Eisenbud and Harris develeoped a theory of limit linear series which explans how (L, V) ...
8
votes
1answer
718 views

what is the connection between D-modules and coordinate bundles?

Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the pro-algebraic group of automorphims of $k[[x_1,...x_n]]$. Let $G_0$ be the subgroup of automorphisms preserving the closed point (note ...
6
votes
3answers
544 views

Generic Noether Normalisation

Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional ...
2
votes
1answer
759 views

Theta Functions and Cousins

So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...
5
votes
2answers
753 views

Can the valuative criteria be checked “on a dense open”?

The valuative criterion for separatedness (resp. properness) says that a noetherian scheme X is separated (resp. proper) if and only if for any DVR R, with fraction field K, any map Spec(K)→...
6
votes
8answers
646 views

What is an example of a function on M_g?

It feels bad talking about a space without knowing a single function on it, hah? So what is a function on the moduli space of curves, from the geometric point of view? From the functorial point of ...
4
votes
4answers
481 views

E_\infty spectrum corresponding to Z_p

First of the questions about derived algebraic geometry from a noobie. The way I understand it, every discrete ring R corresponds to some ring spectrum whose ...
7
votes
2answers
1k views

What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?

Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
12
votes
5answers
2k views

Existence of (smooth) models

Hi everyone, let X be a variety over a field k, S an integral scheme such that the function field K of S is contained in k. An S-scheme X is called model of X/k if X x_S k = X, i.e. if the generic ...
12
votes
3answers
1k views

What is an example of a smooth variety over a finite field F_p which does not embed into a smooth scheme over Z_p?

Such an example of course could not be projective and would not itself lift to Z_p. The context is that one can compute p-adic cohomology of a variety X over a finite field F_p via the cohomology of ...
33
votes
4answers
2k views

Does a scheme have a “separification”?

Background: (1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...
2
votes
3answers
1k views

What is the base change in number theory?

I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change ...
7
votes
1answer
770 views

Why are torsion points dense in an abelian variety?

Hi everyone, let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ of characteristic $p\geqslant 0$. I'm trying to prove that the subgroup $A'$ which is the union of ...
29
votes
4answers
2k views

Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
7
votes
1answer
1k views

Dualizing sheaf on singular curves

I am trying to understand the stabilization map, which takes a prestable curve (a curve with some marked points, and at worst nodal singularities) and returns a stable curve (a curve with some marked ...
7
votes
3answers
2k views

What is the Theorem of the Cube?

What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?
4
votes
2answers
965 views

Non-zero sheaf cohomology

Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
17
votes
2answers
5k views

does a line bundle always have a degree

For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...
11
votes
5answers
2k views

When are Hilbert schemes smooth?

I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a ...
9
votes
5answers
1k views

Compact Kaehler manifolds that are isomorphic as symplectic manifolds but not as complex manifolds (and vice-versa)

What are some examples of compact Kaehler manifolds (or smooth complex projective varieties) that are not isomorphic as complex manifolds (or as varieties), but are isomorphic as symplectic manifolds (...
13
votes
6answers
2k views

Why and how are moduli spaces of (semi)stable vector bundles well-behaved?

The slope of a vector bundle $E$ is defined as $\mu(E) = \deg(E)/\mathrm{rank}(E)$. Then a vector bundle $E$ is called semistable if $\mu(E') \leqslant \mu(E)$ for all proper sub-bundles $E'$. It is ...
11
votes
5answers
2k views

Equivalent Statements of Riemann Hypothesis in the Weil Conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
8
votes
3answers
839 views

How to topologize X(R) when R is a topological ring?

Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...
22
votes
5answers
5k views

Can a quotient ring R/J ever be flat over R?

If R is a ring and J⊂R is an ideal, can R/J ever be a flat R-module? For algebraic geometers, the question is "can a closed immersion ever be flat?" The answer is yes: take J=0. For a less ...
14
votes
3answers
1k views

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
21
votes
2answers
5k views

When is fiber dimension upper semi-continuous?

Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$. When is this function ...
10
votes
5answers
3k views

Ribbon graph decomposition of the moduli space of curves

What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
11
votes
8answers
3k views

Resources on Invariant Theory

Hi, So my question is pretty much summed up by the summary - basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd ...
24
votes
7answers
3k views

How do you see the genus of a curve, just looking at its function field?

Yuhao asked in the 20-questions seminar: The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance. How do you see the genus directly ...
5
votes
2answers
645 views

are deformations of torsion modules always torsion?

Let's say I have a field $\mathbb{K}$ and a flat family of $\mathbb{K}[t]$-modules $M$ over the formal disk $Spec \mathbb{K}[[h]]$. Now, assume that $M/hM$ is torsion as a $\mathbb{K}[t]$-module (...
9
votes
1answer
487 views

Can one check formal smoothness using only one-variable Artin rings?

Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...
16
votes
11answers
7k views

What is the exact statement of “there are 27 lines on a cubic”?

I think there was a theorem, like every cubic hypersurface in $\mathbb P^3$ has 27 lines on it. What is the exact statement and details?
14
votes
1answer
876 views

Is there an example of a scheme X whose reduction X_red is affine but X is not affine?

For Noetherian schemes this follows from Serre's criterion for affineness by a filtration argument.
10
votes
2answers
2k views

What is interesting/useful about Castelnuovo-Mumford regularity?

What is interesting/useful about Castelnuovo-Mumford regularity?
7
votes
1answer
1k views

Can the valuative criteria for separatedness/properness be checked “formally”?

Suppose f:X→Y is a morphism of finite type between locally noetherian schemes. The valuative criterion for separatedness (resp. properness) says roughly that f is a separated (resp. proper) ...
6
votes
1answer
743 views

Stack with affine stabilizers but not quasi-affine diagonal

Give an example of a stack X with affine stabilizer groups and separated but not quasi-affine diagonal. Remarks: 1) If X has finite stabilizer groups then the diagonal is quasi-finite and separated, ...
13
votes
3answers
1k views

Can a coequalizer of schemes fail to be surjective?

Suppose $g,h:Z\to X$ are two morphisms of schemes. Then we say that $f:X\to Y$ is the coequalizer of $g$ and $h$ if the following condition holds: any morphism $t:X\to T$ such that $t\circ g=t\circ h$ ...
24
votes
6answers
5k views

What is the universal property of normalization?

What is the universal property of normalization? I'm looking for an answer something like If X is a scheme and Y→X is its normalization, then the morphism Y→X has property P and any ...