# Questions tagged [ag.algebraic-geometry]

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

16,259
questions

**18**

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### Algebraic varieties which are topological manifolds

Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...

**32**

votes

**2**answers

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### Non-integral scheme having integral local rings

I can show that if $X$ is a scheme such that all local rings $\mathcal{O}_{X,x}$ are integral and such that the underlying topological space is connected and Noetherian, then $X$ is itself integral.
...

**3**

votes

**3**answers

644 views

### Is (relatively) algebraically closed stable under finite field extensions?

Let $F\subset F'$ be a field extension such that $F$ is algebraically closed inside $F'$, i.e. if $x\in F'$ is algebraic over $F$ then $x$ belongs to $F$ itself.
Let now $F\subset L$ be a finite field ...

**14**

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**3**answers

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### Intuition about schemes over a fixed scheme

I am taking a first course on Algebraic Geometry, and I am a little confused at the intuition behind looking at schemes over a fixed scheme. Categorically, I have all the motivation in the world for ...

**2**

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**3**answers

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### Algebraic Varieties which are also Manifolds

Any non-singular projective variety over $\mathbb{C}$ is easily seen to be a smooth manifold. Presumably the same is not true for algebraic varieties - one would not expect varieties with singular ...

**14**

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**5**answers

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### Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension

Let $(A,m_A)$ and $(B,m_B)$ be noetherian local rings and $f:A\rightarrow B$ a local homomorphism. Let $F = B/m_AB$ be the fiber ring and assume that
$$\mathrm{dim}(B) = \mathrm{dim}(A) + \mathrm{dim}...

**27**

votes

**4**answers

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### The Jouanolou trick

In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with ...

**11**

votes

**5**answers

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### Geometry Vs Arithmetic of schemes

Let's suppose we have a Scheme $X$ over the the field $k$, where such a field can be though to be either $\mathbb{C}$ or a finite field $\mathbb{F}_q$. Then having this in mind,
Where do we find some ...

**20**

votes

**1**answer

993 views

### Is Dependent Choice all we really need?

http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...

**2**

votes

**2**answers

232 views

### morphisms from abelian varieties to rational curves.

Let $A$ be an abelian variety and and $\sigma$ an automorphism of $A$. Suppose $f:A\rightarrow P^1$ is a morphism. Is it true that $\sigma$ descends to an automorphism of $P^1$? I seem to remember ...

**12**

votes

**1**answer

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### When is an Albanese variety principally polarized?

Let (X,x) be a pointed projective variety. Then there exists an abelian variety V which is universal for maps of pointed varieties $(X,x) \to (A,e_A)$, called the albanese variety. When X is a curve, ...

**34**

votes

**5**answers

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### Heuristic explanation of why we lose projectives in sheaves.

We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I ...

**16**

votes

**2**answers

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### What does primary decomposition of (sub) modules mean geometrically?

I want to know how I should visualize modules in algebraic geometry. The way we visualize rings, via their spectra, automatically (or by the beauty of its design) depicts primary decomposition of ...

**33**

votes

**1**answer

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### Complex vector bundles that are not holomorphic

Is there an example of a complex bundle on $\mathbb CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We require that the Chern classes of the ...

**3**

votes

**1**answer

670 views

### On algebraic tubular neighbourhoods and Weak Lefschetz

Can one formulate those version of Weak Lefschetz that uses tubular neighbourhoods purely in terms of cohomology of (some) algebraic varieties?
Theorem in 5.1 of Part II in Goresky-MacPherson's "...

**2**

votes

**2**answers

415 views

### What cohomology theories would be interesting for nilpotent cones/nullcones?

As I understand, when we have a nilpotent cone, or a nullcone of a Lie group representation, what seems to be done in a lot of the literature (e.g. Achar&Henderson-"Orbit closures in the enhanced ...

**14**

votes

**7**answers

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### Learning About Schubert Varieties

I am a combinatorist by training and I am interested in learning about the connections between combinatorics and Schubert varieties. The theory of Schubert varieties seems to be a difficult area to ...

**7**

votes

**1**answer

753 views

### Moduli space of flat bundles

Is there a good place to learn about the structure of moduli stack of flat $G$-bundles on an algebraic curve?
Of course, we're just studying representations of a group $\pi_1(X)\to G$ modulo some ...

**41**

votes

**5**answers

4k views

### Open affine subscheme of affine scheme which is not principal

I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme $X$ having an open affine subscheme $U$ which is not principal in $X$? By a principal open of $X ...

**11**

votes

**2**answers

915 views

### Class groups of normal domains over finite fields

Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be ...

**34**

votes

**7**answers

6k views

### Model theoretic applications to algebra and number theory(Iwasawa Theory)

One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...

**14**

votes

**5**answers

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### What is etale descent?

What is etale descent? I have a vague notion that, for example, given a variety $V$ over a number field $K$, etale descent will produce (sometimes) a variety $V'$ over $\mathbb{Q}$ of the same ...

**8**

votes

**3**answers

907 views

### Gerbes for a cyclic group. (or maybe G_m too)

Let μn be the group scheme of n-th roots of unity. If X is a scheme and L is a line bundle on X, then I can construct a μn-gerbe Y over X by letting the S-points of Y be a S-point of X, a line ...

**10**

votes

**2**answers

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### Does a universal Frobenius map exist?

For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p.
Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ...

**8**

votes

**0**answers

420 views

### Lifting sections of bundles

Assume that $X$ is a quasiprojective scheme with a fixed imbedding into a smooth scheme M given by an ideal sheaf $\mathcal{I}$. Assume further that there is a locally free sheaf $E_X$ on $X$ that is ...

**127**

votes

**9**answers

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### Why are flat morphisms “flat?”

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...

**9**

votes

**2**answers

558 views

### When is tensoring with a module representable by a scheme?

Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless ...

**10**

votes

**4**answers

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### Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective object?

X is a Noetherian scheme, F is an injective object in the category of quasi-coherent sheaves on X. U is an open subset of X. Why F's restriction on U is still an injective object in the category of ...

**9**

votes

**1**answer

974 views

### What functor does a Schubert variety represent?

I'm inspired by Yuhao's question. The functor that takes a scheme S to the set of k-dimensional vector subbundles of C^n x S (understanding "subbundle" to mean that the quotient by it is another ...

**11**

votes

**3**answers

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### What functor does Grassmannian represent?

As we know, the Projective space P^n represent the functor sending X to the set of line bundles L on X together with a surjection from the trivial vector bundle to L.
My question is, what functor ...

**15**

votes

**4**answers

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### Is projectiveness a Zariski-local property of modules? (Answered: Yes!)

I know that for a finitely presented $A$-module $M$ ($A$ a commutative ring), TFAE:
1) $M$ is projective;
2) $M$ is max-locally free, meaning that $M_m$ is free for every maximal ideal $m$;
3) $M$ is ...

**36**

votes

**5**answers

3k views

### How to think about CM rings?

There are a few questions about CM rings and depth.
Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to me....

**12**

votes

**2**answers

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### Weil Conjectures for nonprojective algebraic varieties

If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?

**4**

votes

**2**answers

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### Weil Conjectures for Grassmannians

To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?

**12**

votes

**4**answers

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### “Counter”-example for Gauss's Lemma on irreducible polynomials

Gauss's Lemma on irred. polynomial says,
Let R be a UFD and F its field of fractions. If a polynomial f(x) in R[x] is reducible in F[x], then it is reducible in R[x].
In particular, an integral ...

**6**

votes

**1**answer

663 views

### What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be?

What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least ...

**11**

votes

**3**answers

567 views

### Can different modules have the same symmetric algebra? (answered: no)

Algebraic geometry allows one to think of an $A$-module $M$ geometrically as a module of functions on the $A$-scheme $\mathrm{Spec}(\mathrm{Sym}(M))$.
I'm wondering if anything is lost in just ...

**25**

votes

**2**answers

3k views

### What is the algebraic closure of the field with one element?

If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one element.
I saw that the ...

**3**

votes

**1**answer

318 views

### Is the coproduct of fibrant spectra fibrant again?

Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...

**22**

votes

**4**answers

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### What is the relationship between various things called holonomic?

The following things are all called holonomic or holonomy:
A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the ...

**7**

votes

**1**answer

471 views

### Hodge theory for quasi-Kaehler manifolds: where does it break down?

Let $U$ be the complement of a divisor with normal crossings in a smooth compact complex manifold $X$. If $X$ is algebraic, then Deligne describes in "Th\'eorie de Hodge 2" a procedure to equip the ...

**7**

votes

**2**answers

877 views

### Going further on How sections of line bundles rule maps into projective spaces

My question is located in trying to follow the argument bellow.
Given a normal algebraic variety $X$, and a line bundle $\mathcal{L}\rightarrow X$ which is ample, then eventually such a line bundle ...

**6**

votes

**1**answer

514 views

### The 2-sphere and $\mathbb{CP}^1$

As is very well known, the algebraic variety $S^2$ is isomorphic to projective variety $\mathbb{CP}^1$ as a complex manifold. As is also well known, the coordinate ring of $S^2$ is given by $< x,y,...

**11**

votes

**3**answers

895 views

### Equations for Integrable Systems

So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further ...

**22**

votes

**4**answers

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### When are GIT quotients projective?

Some background on GIT
Suppose G is a reductive group acting on a scheme X. We often want to understand the quotient X/G. For example, X might be some parameter space (like the space of possible ...

**0**

votes

**2**answers

1k views

### algebraic equivalence of divisors

for the embedding defined by very ample divisors, if they are lin. eq, then the embeddings are the "same" (up to a linear transformation). What do we know if given that the divisors are algebraically ...

**1**

vote

**1**answer

97 views

### Is the total space of a module connected?

Let $A$ be a ring and $E$ a module. If $\mathrm{spec} A$ is connected, then so is $\mathrm{spec} S^\bullet E$. If this is not true in general, then what are some minimal conditions that make it true?

**10**

votes

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### Counting points on varieties of low codimension

The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...

**11**

votes

**2**answers

594 views

### When can cohomology be calculated on the coarse moduli space?

Suppose $\cal{X}$ is a DM-stack, and X its coarse moduli space. Let F be a sheaf on $\cal{X}$, and $\pi : \mathcal{X} \to X$ the projection. In all examples I have seen, it has been true that
$H^i(\...

**23**

votes

**2**answers

1k views

### Least number of non-zero coefficients to describe a degree n polynomial

I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it.
Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be ...