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8 votes
2 answers
823 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 ...
David E Speyer's user avatar
9 votes
1 answer
841 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) ...
Matt Satriano's user avatar
8 votes
1 answer
1k 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 ...
Ian Shipman's user avatar
  • 1,038
8 votes
3 answers
921 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 ...
Simon Wadsley's user avatar
2 votes
1 answer
927 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 ...
Chris Schommer-Pries's user avatar
5 votes
3 answers
1k 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)→...
Anton Geraschenko's user avatar
7 votes
8 answers
747 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 ...
Yuhao Huang's user avatar
  • 5,052
5 votes
4 answers
826 views

$E_\infty$ spectrum corresponding to $\Bbb Z_p$

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 $\pi_0$ is $R$. Now consider $p$-adic ...
Ilya Nikokoshev's user avatar
9 votes
2 answers
2k 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.
David Zureick-Brown's user avatar
15 votes
5 answers
3k 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 ...
Lars's user avatar
  • 4,450
15 votes
2 answers
2k 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 ...
David Zureick-Brown's user avatar
39 votes
4 answers
3k 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 ...
Anton Geraschenko's user avatar
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 ...
Ilya Nikokoshev's user avatar
10 votes
1 answer
1k 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 ...
Lars's user avatar
  • 4,450
32 votes
4 answers
3k 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 ...
Ilya Nikokoshev's user avatar
7 votes
1 answer
2k 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 ...
Kevin H. Lin's user avatar
13 votes
3 answers
3k 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?
Chris Schommer-Pries's user avatar
7 votes
2 answers
1k 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?...
Georges Elencwajg's user avatar
20 votes
2 answers
10k 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 ...
solbap's user avatar
  • 3,968
14 votes
5 answers
4k 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 ...
Kevin H. Lin's user avatar
10 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 (...
Kevin H. Lin's user avatar
19 votes
6 answers
4k 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 ...
Kevin H. Lin's user avatar
20 votes
5 answers
4k views

Equivalent statements of the 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 ...
Brandon Levin's user avatar
13 votes
3 answers
1k 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 ...
Brandon Levin's user avatar
28 votes
5 answers
9k views

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

If $R$ is a ring and $J\subset R$ 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=...
Anton Geraschenko's user avatar
18 votes
3 answers
2k 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?
David Zureick-Brown's user avatar
30 votes
2 answers
10k 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 ...
Anton Geraschenko's user avatar
11 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?
Kevin H. Lin's user avatar
20 votes
10 answers
7k views

Resources on invariant theory

What are resources on invariant theory? 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 prefer online / freeish ...
27 votes
7 answers
4k 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 ...
20 questions's user avatar
  • 1,059
6 votes
2 answers
673 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 (...
Ben Webster's user avatar
  • 44.7k
9 votes
1 answer
531 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 ...
David Zureick-Brown's user avatar
22 votes
11 answers
13k 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?
Ilya Nikokoshev's user avatar
15 votes
1 answer
2k 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.
David Zureick-Brown's user avatar
20 votes
4 answers
4k views

What is interesting/useful about Castelnuovo-Mumford regularity?

What is interesting/useful about Castelnuovo-Mumford regularity?
David Zureick-Brown's user avatar
10 votes
1 answer
2k 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) ...
Anton Geraschenko's user avatar
9 votes
1 answer
1k 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, ...
David Rydh's user avatar
  • 5,039
19 votes
3 answers
2k 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 $t\circ g=t\circ h$ ...
Anton Geraschenko's user avatar
32 votes
6 answers
9k 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 ...
Anton Geraschenko's user avatar
7 votes
3 answers
585 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 ...
solbap's user avatar
  • 3,968
9 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. ...
Anton Geraschenko's user avatar
12 votes
2 answers
2k views

Non-quasi separated morphisms

What are some examples of morphisms of schemes which are not quasi separated?
Matt Satriano's user avatar
2 votes
1 answer
406 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.
David Zureick-Brown's user avatar
3 votes
2 answers
857 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 ...
David Zureick-Brown's user avatar
38 votes
18 answers
24k 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
2k views

Ignore this question [closed]

This question is a hacky way to create some tags for you to use. Move along.
Anton Geraschenko's user avatar

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