Questions tagged [ag.algebraic-geometry]

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

Filter by
Sorted by
Tagged with
18
votes
3answers
2k views

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
2answers
3k views

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
3answers
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
votes
3answers
3k views

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
votes
3answers
3k views

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
votes
5answers
3k views

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
4answers
4k views

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
5answers
2k views

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
1answer
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
2answers
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
1answer
1k views

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
5answers
3k views

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
2answers
2k views

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
1answer
2k views

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
1answer
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
2answers
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
7answers
2k views

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
1answer
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
5answers
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
2answers
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
7answers
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
5answers
3k views

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
3answers
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
2answers
1k views

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
0answers
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
9answers
13k views

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
2answers
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
4answers
2k views

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
1answer
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
3answers
2k views

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
4answers
2k views

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
5answers
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
2answers
1k views

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
2answers
1k views

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
4answers
5k views

“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
1answer
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
3answers
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
2answers
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
1answer
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
4answers
1k views

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
1answer
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
2answers
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
1answer
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
3answers
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
4answers
3k views

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
2answers
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
1answer
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
2answers
381 views

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
2answers
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
2answers
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 ...

1
318 319
320
321 322
326