# Questions tagged [ag.algebraic-geometry]

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

16,277
questions

**8**

votes

**4**answers

2k views

### Smoothness of Symmetric Powers

Here's something that's been bothering me, and that's come up again for me recently while reading some stuff about Hilbert schemes of points (Nakajima's lectures, specifically):
Let $C$ be an ...

**14**

votes

**6**answers

2k views

### “Every scheme as a sheaf” references?

I have sometimes hard time reading papers that are written in the language of schemes being replaced by the functors they represent (I have especially homotopy scheme theory in mind).
I think the ...

**4**

votes

**1**answer

542 views

### Moduli spaces of coherent sheaves on K3s

Reading 2007 paper A tour of theta dualities on moduli spaces of sheaves by Alina Marian and Dragos Oprea.
Why is any moduli space of coherent sheaves on a K3 surface deformation equivalent to a ...

**12**

votes

**3**answers

685 views

### Schemes of Representations of Groups

Let $G$ be a group, say finitely presented as $\langle x_1,\ldots,x_k|r_1,\ldots,r_\ell\rangle$. Fix $n\geq 1$ a natural number. Then there exists a scheme $V_G(n)$ contained in $GL(n)^k$ given by ...

**2**

votes

**1**answer

237 views

### subspace topology for functors

let $X : Ring \to Set$ be a functor (a Z-functor in the language of demazure, gabriel) and $V \subseteq X$ a locally closed subfunctor. assume that $U \subseteq V$ is an open subfunctor. does then ...

**5**

votes

**0**answers

344 views

### ring-valued points of locally ringed spaces

of course, one should expect that the concept of ring-valued points is not well-behaved for locally ringed spaces (LRS). I want to see examples for this.
so consider $LRS \to Set^{Ring}, X \mapsto X(-...

**10**

votes

**3**answers

2k views

### on chern classes and Riemann Roch theorem for torsion-free sheaves on singular (possibly multiple) curve

I'm looking for a definition of Chern class (at least the first one) for a torsion-free sheaf $F$ (not necessarily locally free) on a singular curve (for simplicity can assume all the singularities ...

**11**

votes

**1**answer

708 views

### A GAGA question

A GAGA question.
Say I have a ``quasi-projective'' (*) subvariety X over the complex
numbers within a smooth complex algebraic variety Z.
True or False: The analytic and algebraic closure ...

**14**

votes

**2**answers

671 views

### A local-to-global principle for being a rational surface

Let $k$ be a number field and $F$ a $1$-variable function field over $k$ (a finitely generated extension of $k$, of transcendence degree $1$, in which $k$ is algebraically closed). If $F$ becomes the ...

**3**

votes

**1**answer

226 views

### GW invariants for varieties with negative first Chern class

Does there exist any theorem claiming that if a variety with negative first Chern class has no rational curves then every GW invariant is zero?

**16**

votes

**1**answer

828 views

### Coarse moduli spaces over Z and F_p

I would like to know to what extent it is possible to compare fibers over $\mathbb{F}_p$ of coarse moduli spaces over $\mathbb{Z}$, and coarse moduli spaces over $\mathbb{F}_p$. I ask a more precise ...

**42**

votes

**5**answers

6k views

### Colimits of schemes

This is related to another question.
I've found many remarks that the category of schemes is not cocomplete. The category of locally ringed spaces is cocomplete, and in some special cases this turns ...

**5**

votes

**0**answers

603 views

### Analysis analogue of Orlov's theorem?

Mukai's theorem states that if $X$ is an abelian variety, and $\check{X}$ is the dual abelian variety, then the Fourier-Mukai transform corresponding to the Poincare line bundle on $X \times \check{X}$...

**34**

votes

**7**answers

6k views

### Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?

**2**

votes

**2**answers

434 views

### tamely branched cover over P^1

k is an algebraically closed field, X is a smooth, connected, projective curve over k. f: X-->P^1 is a finite morphism. Let t be a parameter of P^1, suppose f is etale outside t=0 and t=\infty, and ...

**21**

votes

**1**answer

2k views

### Is every smooth affine curve isomorphic to a smooth affine plane curve?

As suggested by Poonen in a comment to an answer of his question about the birationality of any curve with a smooth affine plane curve we ask the following questions:
Q) Is it true that every smooth ...

**1**

vote

**3**answers

780 views

### On the Clifford index of a curve

Let X be an algebraic curve and c be the Clifford index of X.
When c is small (e.g c=1), what is the classification of the line bundle who computes c?

**29**

votes

**3**answers

4k views

### Matrix factorizations and physics

I have heard during some seminar talks that there are applications of the theory of
matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...

**18**

votes

**3**answers

963 views

### What is the Zariski closure of the space of semisimple Lie algebras?

Given Leonid Positselski's excellent answer and comments to this question, I expect that the present one is a hard question. Recall that the Lie algebra structures on a (finite-dimensional over $\...

**12**

votes

**1**answer

356 views

### Why are non-singleton covering families often ignored?

It seems to me that frequently when discussing stack conditions and descent, people consider only singleton covering families, i.e. there is some single covering map $U\to X$, for which one constructs ...

**4**

votes

**3**answers

1k views

### Degree of an embedded algebraic variety

Let $X$ be an algebraic variety and $A$ is a ample divisor on $X$. Let $m$ be a sufficiently large natural number such that $X \overset{\varphi_{mA}}{\to} \mathbf{P}H^0(X, \mathcal{O}_X(mA))$ defined ...

**2**

votes

**1**answer

670 views

### A strange logical implication in algebraic geometry

So there's an old theorem of Lang and Weil showing that the Riemann hypothesis for curves over finite fields implies a kind of quasi-riemann hypothesis for surfaces over finite fields.
I am wondering:...

**10**

votes

**3**answers

1k views

### Rings of integers of function fields

This might be a somewhat silly and inconsequential question, but it's aroused my curiosity. One has the theorem in commutative algebra that the integral closure of a domain $A$ in its field of ...

**54**

votes

**3**answers

3k views

### Is “semisimple” a dense condition among Lie algebras?

The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some ...

**25**

votes

**1**answer

2k views

### Is every curve birational to a smooth affine plane curve?

Is every curve over $\mathbf{C}$ birational to a smooth affine plane curve?
Bonnie Huggins asked me this question back in 2003, but neither I nor the few people I passed it on to were able to answer ...

**4**

votes

**2**answers

646 views

### Convergence of quantum cohomology

For which manifolds or varieties is quantum cohomology known to converge? Are there any manifolds for which quantum cohomology is known to not converge? I seem to have the impression that quantum ...

**1**

vote

**1**answer

283 views

### Singular matrix and wedge product

If I have a singular matrix $X$ with components $X_{\mu\nu}$:
$t^{\nu}X_{\mu\nu}=0$
By considering now $X_{\mu\nu}$'s as components of a 2-form can I say that:
$X\wedge X=0$ ?
If yes, how?

**3**

votes

**1**answer

310 views

### “Eigenvalue characters”

This question is an addition to my question on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group ...

**16**

votes

**1**answer

2k views

### Eichler-Shimura isomorphism and mixed Hodge theory

Let $Y(N),N>2$ be the quotient of the upper half-plane by $\Gamma(N)$ (which is formed by the elements of $SL(2,\mathbf{Z})$ congruent to $I$ mod $N$). Let $V_k$ be the $k$-th symmetric power of ...

**61**

votes

**1**answer

5k views

### Smooth proper scheme over Z

Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- ...

**20**

votes

**2**answers

1k views

### Canonical topology on the category of schemes?

Every category admits a Grothendieck topology, called canonical, which is the finest topology which makes representable functor into sheaves.
Is there a concrete description of the canonical topology ...

**11**

votes

**3**answers

3k views

### References for Donaldson-Thomas theory and Pandharipande-Thomas theory?

I'm looking for good introductory references for Donaldson-Thomas theory and Pandharipande-Thomas theory. I know a bit about Gromov-Witten theory, but almost nothing about Donaldson-Thomas and ...

**12**

votes

**3**answers

1k views

### How does one find vanishing algebraic cycles?

I have a question, related to what I asked before.
Let's consider a smooth hyperplane section $X$ of a smooth projective variety $Y$ over $\mathbb C$.
According to Weak Lefschetz theorem, cohomology ...

**26**

votes

**9**answers

4k views

### Do there exist modern expositions of Klein's Icosahedron?

Reading Serre's letter to Gray
, I wonder if now modern expositions of the themes in Klein's book
exist. Do you know any?

**12**

votes

**5**answers

4k views

### reduced ⊗ reduced = reduced; what about connected?

Several questions actually.
All rings and algebras are supposed to be commutative and with $1$ here.
(1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and $...

**3**

votes

**3**answers

1k views

### resolution of singularities on surfaces

Let |V| be a (incomplete) linear series on a nonsingular projective surface. Hironaka says that there is a resolution of the singularities of |V| along smooth centers. If the base locus of |V| is just ...

**19**

votes

**7**answers

2k views

### Higher Dimensional Gromov-Witten Theories

So, a basic set-up in modern enumerative geometry is that you have some object $X$ (say, a "nice" stack, for whatever definition of "nice" you need) and then you want to "count" the curve of genus $g$ ...

**6**

votes

**3**answers

1k views

### What's the classification of the algebraic subgroups of Sp(4,R)?

Hi! I would like to know if there is an explicit classification of the algebraic (i.e., Zariski closed) subgroups of the symplectic group Sp(4,R) and/or more generally Sp(2n,R) somewhere in the ...

**2**

votes

**2**answers

464 views

### Real spectrum of ring of continuous semialgebraic functions

Let R be a real closed field, and let U be a semialgebraic subset of $R^n$. Let $S^0(U)$ be the ring of continuous R-valued semialgebraic functions. Also let $\tilde{U}$ be the subset of Spec$_r (R[...

**3**

votes

**1**answer

883 views

### plane hyperelliptic curves

I'm aware that h/w problems are frowned upon (understandably) here. However - this really is just inspired by some h/w related confusion, so hopefully that's ok.
Anyway, can one have a smooth ...

**2**

votes

**2**answers

636 views

### Curves on elliptic ruled surfaces?

Let $S\overset{\pi}{\to} E$ be a ruled surface over an elliptic curve over complex field. Clearly, there are rational curves and elliptic curves on $S$. Is there any higher genus curves on $S$. Are ...

**2**

votes

**1**answer

150 views

### Lower bound for characteristic variety

Let K be an algebraically closed field of char. 0, let A_n(K) be the Weyl algebra. Let I in A_n(K) be a left ideal generated by p elements. Set M := A_n / I.
Does the following then hold?
dim Ch(M) \...

**50**

votes

**6**answers

4k views

### What do Weierstrass points look like?

As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...

**11**

votes

**4**answers

846 views

### Geometry of the multilagrangian Grassmannian

Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...

**2**

votes

**1**answer

256 views

### Descend finite etale algebras

Let $\pi:X\to\mathcal X$ be a presentation of an Artin stack $\mathcal X$ of finite type over a field $k,$ and let $f:Y\to X$ be a finite \'etale covering. Does there exist a finite \'etale covering $...

**38**

votes

**6**answers

4k views

### Arbitrary products of schemes don't exist, do they?

Thinking of arbitrary tensor products of rings, $A=\otimes_i A_i$ ($i\in I$, an arbitrary index set), I have recently realized that $Spec(A)$ should be the product of the schemes $Spec(A_i)$, a ...

**8**

votes

**4**answers

765 views

### Algebraic geometry for cocommutative corings with counit.

Is there a notion of algebraic geometry for these objects? If we take the dual category of the category of cocommutative corings with counit, is there geometry in it in a sense dual to affine schemes? ...

**6**

votes

**1**answer

538 views

### Detecting etale maps on reduced points

Suppose I have a morphism of schemes for which I know the relative cotangent complex is trivial, and the map on reduced subschemes is an isomorphism. Is the map an isomorphism?
More generally, given ...

**19**

votes

**3**answers

2k views

### A good example of a curve for geometric Langlands

I'm currently working through Frenkel's beautiful paper:
http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf.
I'm looking for a good example of a projective curve to get my hands dirty, and go ...

**10**

votes

**7**answers

2k views

### Is there any rational curve on an Abelian variety?

Is that true that there is no rational curves contained in an Abelian variety? If it's true, is that because abelian varieties are not uniruled? How do I know whether an abelian variety is not ...