# Questions tagged [ag.algebraic-geometry]

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

16,478
questions

**17**

votes

**2**answers

6k 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 ...

**8**

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 (...

**14**

votes

**6**answers

3k 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 ...

**12**

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 ...

**9**

votes

**3**answers

892 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 ...

**23**

votes

**5**answers

6k 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 ...

**15**

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?

**25**

votes

**2**answers

6k 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

4k 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

653 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

493 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 ...

**18**

votes

**11**answers

8k 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

991 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.

**13**

votes

**2**answers

3k 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

810 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 $t\circ g=t\circ h$ ...

**25**

votes

**6**answers

6k 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 ...

**5**

votes

**3**answers

485 views

### Weil divisors on non Noetherian schemes

Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...

**8**

votes

**3**answers

1k views

### If $\Omega_{X/Y}$ is locally free of rank $\mathrm{dim}\left(X\right)-\mathrm{dim}\left(Y\right)$, is $X\rightarrow Y$ smooth?

Suppose I have a morphism $f:X\rightarrow Y$ such that the relative sheaf of differentials $\Omega_{X/Y}$ is locally free. Does it follow that $f$ is smooth?
The answer is no, but for a silly reason. ...

**9**

votes

**2**answers

2k views

### Non-quasi separated morphisms

What are some examples of morphisms of schemes which are not quasi separated?

**2**

votes

**1**answer

337 views

### Are there any criteria for a presheaf which is an etale sheaf to be a sheaf in the fppf topology?

I am happy to hear answers to variants too. For instance, my situation I actually have a sheaf in the smooth topology.

**3**

votes

**2**answers

630 views

### Is there an example of an algebraic stack whose closed points have affine stabilizers but whose diagonal is not affine?

Burt Totaro has a result that for a certain class of algebraic stacks, having affine diagonal is equivalent to the stabilizers at closed points begin affine. Is there an example of this equivalence ...

**28**

votes

**18**answers

17k views

### Learning about Lie groups

Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.

**0**

votes

**0**answers

1k views

### Ignore this question [closed]

This question is a hacky way to create some tags for you to use. Move along.