# Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

17,035
questions

**7**

votes

**2**answers

251 views

### What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)

Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...

**43**

votes

**8**answers

5k views

### When are there enough projective sheaves on a space X?

This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...

**14**

votes

**2**answers

475 views

### Algebraic data and purity associated to codimension greater than 2

Consider the following statement: Let $X$ be a smooth and geometrically integral variety over a field $k$ and let $U$ be any open subset of $X$ whose complement is of codimension greater or equal to $...

**35**

votes

**9**answers

4k views

### What is a deformation of a category?

I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, ...

**25**

votes

**5**answers

3k views

### Flips in the Minimal Model Program

In order get a minimal model for a given a variety $X$, we can carry out a sequence of contractions $X\rightarrow X_1\ldots \rightarrow X_n$ in such a way that that every map contracts some curves on ...

**4**

votes

**1**answer

801 views

### A working generalization of Weil divisors

Hartshorne defines Weil divisors under the hypotheses "Noetherian integral separated scheme regular in codimension 1", which, for example, ensures that the divisor of a rational function is a finite ...

**34**

votes

**7**answers

7k views

### Geometric meaning of the Euler sequence on $\mathbb{P}^n$ (Example 8.20.1 in Ch II of Hartshorne)

Is there any geometric way to understand the exact sequence in Example 8.20.1 in Ch II of Hartshorne (p. 182), or its dual from theorem 8.13?
Here is the sequence:
$0\to O_{\mathbb{P}^n}\to O_{\...

**6**

votes

**2**answers

679 views

### Do quotients of representable sheaves represent quotients?

Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...

**25**

votes

**5**answers

4k views

### Logic comment in Mumford's Red Book

In Mumford's "The red book of varieties and schemes" one of the examples (G on pg 74) is the space Spec $(\prod_{i=1}^\infty k)$, where $k$ is a field. He mentions that "Logicians assure us that we ...

**37**

votes

**10**answers

5k views

### Algebraic Geometry versus Complex Geometry

This question is motivated by this one.
I would like to hear about results concerning complex projective varieties which
have a complex analytic proof but no known algebraic proof; or
have an ...

**27**

votes

**5**answers

4k views

### Pushouts in the Category of Schemes

When does it make sense to glue schemes together along subschemes?
In particular: is there a way to glue two schemes together along a closed point (say we're working over a field)? Can you glue two ...

**7**

votes

**2**answers

873 views

### Richardson varieties over finite fields

Let me start with some background to set the notation before I ask my question.
Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel ...

**12**

votes

**0**answers

571 views

### Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree
in $\mathbb C^{n+1})$.
The fiber over $(\lambda:\mu) \in ...

**2**

votes

**5**answers

973 views

### Reference for explicit calculation of blowups (of ideals) and strict transforms

Can one suggest some references where explicit calculations for blow up technique(along an ideal) and strict transformation is done in different examples?

**3**

votes

**2**answers

2k views

### Modules, Sheaves and Vector bundles

Given a graded ring $S$ and a graded S-module $M$ we can carry out a construction in order to get $\tilde{M}$, which is a sheaf over the scheme $\mathrm{Proj}~ S$. With this in view, I have an ...

**10**

votes

**2**answers

729 views

### Exotic automorphisms of the fundamental group of a curve?

A while back, Jordan S. Ellenberg brought the following problem to my attention.
If $G$ is a residually finite group, let $\widehat G$ be its profinite completion.
Let $S$ be a closed surface of ...

**12**

votes

**3**answers

3k views

### What is the relationship between algebraic geometry and quantum mechanics?

The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...

**31**

votes

**5**answers

8k views

### The Relationship between Complex and Algebraic Geomety

I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one ...

**9**

votes

**1**answer

388 views

### Existence of hyperelliptic curve with specific number of points in a family

Hi,
the following question was posed to me, it apparently has applications for linear codes. Let n>1, and $K = \rm{GF}(2^n)$. Let $k$ be coprime to $2^n-1$. Does there always exist $a \neq 0$ in $K$ ...

**1**

vote

**3**answers

2k views

### Various Cartan's Lemmata

I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...

**7**

votes

**1**answer

253 views

**36**

votes

**3**answers

4k views

### What do higher Chow groups mean?

Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...

**5**

votes

**2**answers

861 views

### Singular K3 — mathematical meaning?

There's a very interesting text by Cumrun Vafa called Geometric Physics.
Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration:
...

**15**

votes

**4**answers

1k views

### K3 surfaces with good reduction away from finitely many places

Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...

**11**

votes

**3**answers

1k views

### level sets of multivariate polynomials

Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...

**16**

votes

**3**answers

2k views

### The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.

The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points.
One high dimensional extension ...

**16**

votes

**8**answers

3k views

### Hironaka desingularisation theorem — new proofs in literature?

I'm wondering what the landscape looks like for proofs of Hironaka's desingularisation theorem.
Are there many proofs in the literature?
Is there a commonly accepted simplest bare-knuckle proof ...

**10**

votes

**1**answer

455 views

### Can an algebraic space fail to have a unviersal map to a scheme?

Let $\mathcal{X}$ be an algebraic space. Can it happen that there does not exist a map $\mathcal{X} \to X$ with $X$ a scheme that is initial for maps from $\mathcal{X}$ to schemes? Are there ...

**7**

votes

**1**answer

973 views

### Valuative criterion for properness

Let $f : X \rightarrow Y$ be a finite type morphism of Noetherian schemes. The valuative criterion for properness runs as follows. Suppose that for any DVR $R$ with fraction field $K$ that any $K$-...

**19**

votes

**3**answers

3k views

### When is an algebraic space a scheme?

Sometimes general theory is "good" at showing that a functor is representable by an algebraic spaces (e.g., Hilbert functors, Picard functors, coarse moduli spaces, etc). What sort of general ...

**21**

votes

**5**answers

6k views

### Maps to projective space determined by a line bundle

The following should be pretty standard for any algebraic geometer.
Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for ...

**6**

votes

**1**answer

833 views

### Uniformization in algebraic/arithmetic geometry?

Jonah's question makes me wonder: What is with uniformization in algebraic/arithmetic geometry? E.g. this article by Faltings seems to be about that, the Shimura-Taniyama statement too, Mochizuki ...

**2**

votes

**4**answers

524 views

### a question on function fields (extending my previous question)

Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting ...

**15**

votes

**5**answers

2k views

### Can we count isogeny classes of abelian varieties?

Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...

**6**

votes

**3**answers

1k views

### Examples of birational equivalence of a variety and a hypersurface

There's an algebraic geometry theorem (I.4.9 in Hartshorne) that says: any variety of dimension r (over an algebraically closed field) is birationally equivalent to a hypersurface in projective space ...

**2**

votes

**1**answer

278 views

### k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?

If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and ...

**34**

votes

**4**answers

3k views

### How should one approach tropical mathematics?

Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across this paper which presented an idea that struck me as really remarkable....

**87**

votes

**10**answers

9k views

### equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...

**42**

votes

**1**answer

11k views

### Consequences of Geometric Langlands

So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...

**43**

votes

**6**answers

4k views

### Universal definition of tangent spaces (for schemes and manifolds)

Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the ...

**12**

votes

**3**answers

3k views

### Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...

**4**

votes

**1**answer

211 views

### Comparing maps of reduced schemes

Nice fact:
Suppose f:X->Y is a map of schemes and Z⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and Z are reduced, then it follows that f factors through Z. ...

**10**

votes

**1**answer

641 views

### a question on function fields

Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that ...

**4**

votes

**3**answers

573 views

### Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?

By the GAGA principle we know that a holomorphic vector bundle E->X is analitically isomorphic to an algebraic one, say F->X, and by definition F is locally trivial in the Zariski topology. But since ...

**19**

votes

**2**answers

1k views

### Is Hodge theory somehow connected with a Galois group action Gal(C/R)?

I'm currently taking a course in Hodge theory ... and I wonder if all the splittings in $\{i,-i\}$ Eigenvalue pairs come from the Galois group action (of the extension $\mathbb{R}\rightarrow\mathbb{C}$...

**53**

votes

**8**answers

6k views

### Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...

**9**

votes

**3**answers

1k views

### Crepant resolutions of toric varieties

Given a toric variety, is it easy to see if a crepant resolution exists? If so, how can it be explicitly constructed?

**3**

votes

**1**answer

1k views

### Iso-lines to 3D Surface Generation

I have a set of isolines points ( or contour points) such as this:
Each point has their own respective X, Y and Z. Since they are isolines, that means that all of the points will have a unique X-Y ...

**13**

votes

**2**answers

780 views

### Can Hom_gp(G,H) fail to be representable for affine algebraic groups?

Let $G$ and $H$ be affine algebraic groups over a scheme $S$ of characteristic 0 and let $\textbf{Hom}_{S,gp}(G,H)$ be the functor $T \mapsto \text{Hom}\_{T,gp}(G,H)$
Theorem (SGA 3, expose XXIV, 7....

**13**

votes

**12**answers

8k views

### Are there any interesting connections between Game Theory and Algebraic Topology?

I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? ...