# Questions tagged [ag.algebraic-geometry]

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

14,707 questions

**30**

votes

**6**answers

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

**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 ...

**21**

votes

**6**answers

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

**1**answer

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

**2**answers

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

**8**answers

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### 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

**3**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**5**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**1**answer

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

**2**answers

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

**8**answers

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

**4**answers

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

**2**answers

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

**5**answers

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

**3**answers

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

**4**answers

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

**3**answers

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

**1**answer

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

**4**answers

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

**1**answer

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

**3**answers

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

**2**answers

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

**2**answers

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

**5**answers

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

**5**answers

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

**6**answers

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

**5**answers

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

**3**answers

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

**5**answers

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

**3**answers

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

**2**answers

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

**5**answers

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

**8**answers

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

**7**answers

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

**2**answers

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

**1**answer

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

**11**answers

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

**1**answer

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

**2**answers

2k views

### What is interesting/useful about Castelnuovo-Mumford regularity?

What is interesting/useful about Castelnuovo-Mumford regularity?

**7**

votes

**1**answer

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

**1**answer

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

**3**answers

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

**6**answers

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 ...