All Questions
Tagged with gn.general-topology ag.algebraic-geometry
117
questions
15
votes
3
answers
1k
views
Is symmetric power of a manifold a manifold?
A Hausdorff, second-countable space $M$ is called a topological manifold if $M$ is locally Euclidean. Let $SP^{n}(M): = \left(M \times M \times \cdots \times M \right)/ \Sigma_{m}$, where product is ...
- 339
0
votes
0
answers
69
views
Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
- 227
3
votes
1
answer
113
views
Spectrum of continuous functions as a semigroup
Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the ...
0
votes
0
answers
135
views
Topological property of an algebraic stack and its presentation
I started to learn algebraic stacks this January. I found there are several properties of algebraic stacks which are defined in terms of their underlying topological spaces, for example, connectedness,...
- 361
2
votes
0
answers
237
views
Blow up at an ordinary double point
Let $X \subset \mathbb{C}^n$ be a complex complete intersection surface with only ordinary double point singularities. Let $o$ be such an ordinary double point.
Let $\tilde{X}$ be the strict transform ...
1
vote
0
answers
44
views
Is there some kind of construction of a "canonical unirational variety" like the one for toric varieties?
Toric varieties in some sense a "canonical rational variety" in that one can construct them from purely combinatorial data and this combinatorial data makes it possible to turn many ...
- 759
1
vote
0
answers
57
views
Space of valuations is spectral space and what does it mean to say that conditions are closed conditions
I am reading lecture 3 of Conrad notes (link : https://math.stanford.edu/~conrad/Perfseminar/ ), in which he proves space of valuations is a spectral space. Last theorem of lecture 3.
We have a map $j:...
- 11
2
votes
0
answers
66
views
a connected geometrically unibranch algebraic stack of finite type over a field is irreducible
Let $f:X\to \mathfrak{X}$ be a smooth presentation of geometrically unibranch connected algebraic stack by a scheme, which is geometrically unibranch since being geom. unibranch is local in smooth ...
- 361
2
votes
1
answer
323
views
$4$-manifold with simply connected boundary
This may be a very silly question but I could not get any counter-example.
Let $M$ be a compact differential $4$-manifold with boundary $dM$.
Suppose that the inclusion map induced map $\pi_1(dM) \to \...
- 1,039
2
votes
1
answer
208
views
Formula for the Euler characteristic of a local system on $\mathbb{P}^1$
Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
- 6,567
3
votes
0
answers
106
views
Two topologies on the space of maps from an algebraically closed field to a projective variety
This question is related to this one but I have written this in a self-contained manner.
All varieties are complex varieties.
For quasi-projective variety $U$ and a projective variety $X$ we can ...
- 5,791
3
votes
0
answers
101
views
Space of algebraic maps and quotient under finite group action
For a normal (you can assume smooth for this problem) quasi projective complex variety $X$ and a projective complex variety $Y$, we can endow the space of the set of morphisms $\operatorname{Mor}(X,Y)$...
- 5,791
1
vote
1
answer
171
views
Is the image of a constructible set between Jacobson spaces constructible if the map takes closed points to closed points?
Let $X$ and $Y$ be two spectral Jacobson spaces and let $f: X \to Y$ be a spectral morphism, i.e. $f$ is continuous and the inverse image of a quasi-compact open is quasi-compact. Suppose further that ...
4
votes
0
answers
257
views
Is the "naive" version of Chevalley's theorem still true?
Reposting from math.se in case more people are interested here.
Chevalley's theorem says that if $f \colon X \to Y$ is a morphism of finite presentation of schemes and $C \subset X$ is constructible, ...
- 532
1
vote
0
answers
200
views
Question regarding affine fibre bundles
Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
- 585
5
votes
1
answer
400
views
Ring of continuous functions is a Jacobson ring
Let $X$ be an infinite discrete topological space. Is $$C_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
1
vote
1
answer
189
views
Approximations by compact sub-spaces
Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim_{a\in J} K_a$$
for $J$ a directed set ...
user335418
8
votes
1
answer
425
views
Let $X$ be a manifold. Is it true that $\beta X\cong \operatorname{Specm}(C^\infty(X))$?
Let $X$ be a (smooth) manifold. It's well known that its Stone-Cech compactification $\beta X$ is homeomorphic to $\operatorname{Specm}(C(X))$, with its Zariski topology.
Is $\beta X$ also ...
- 1,054
12
votes
3
answers
1k
views
A quotient space of complex projective space
Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...
- 321
2
votes
0
answers
123
views
(Local) simple connectedness of irreducible algebraic varieties
Let $\mathbb k$ be an algebraically closed field of characteristic zero.
I have two questions:
(1) Is an irreducible algebraic variety $X/\mathbb k$ of dimension at least 2 locally simply connected?
(...
- 129
1
vote
0
answers
79
views
Homotopy type of a general fiber for a polynomial
Define a polynomial $f: \mathbb{C}^2 \to \mathbb{C}$ by $f(x,y)= x(x(2y+1)+1)(x(2y+1)-1).$ The inverse image of zero (i.e. $f^{-1}(0)$) is $\mathbb{C}\cup \mathbb{C}^*\cup \mathbb{C}^*$ (the unions ...
- 1,039
1
vote
0
answers
111
views
Inclusion inducing isomorphism at all level except one
Let $V$ is a projective hypersurface of dimension $3$ and $D$ be divisor at infinity of $V$ (assume $D$ has isolated singularities). It is known that the third homology of both $V$ and $D$ are hard to ...
- 1,039
1
vote
0
answers
58
views
Topological intuition for the cancellation property of separated maps w.r.t a class of properties of continuous maps
Recall a continuous map is separated if its diagonal is closed. This is equivalent to the fibers being relatively Hausdorff in the total space.
Proposition. Suppose $\mathrm P$ is a class of ...
- 10.3k
3
votes
0
answers
211
views
Products of projective spaces
This is a question about projective spaces which is either well known or totally misconceived, and it would be nice to know which. It arose from looking at the pure state spaces on finite dimensional $...
- 1,471
3
votes
1
answer
648
views
Motives and topological data analysis
Here is some meta mathematics question.
During the last decade there has been some progress in the field of applied maths, called topological data analysis.
The setup starts with some set of points in ...
- 1,397
5
votes
1
answer
176
views
Nonvanishing section of infinite-dimensional tautological bundle II
This is a follow-up question on a previous question of mine, which ended up to be trivial, because I overlooked the obvious problem with Hilbert space bundles, which I fix here.
Let us write $E$ for ...
- 9,481
8
votes
0
answers
501
views
When is a constructible set locally closed?
Let $X$ be a topological space (or more specifically, $\mathbb{C}^N$ endowed with the Zariski topology), and let $S \subseteq X$ be a constructible set, i.e. $S=\cup_{i=1}^n C_i \cap U_i$, where the $...
- 1,010
13
votes
0
answers
304
views
Do connected algebraic stacks have a smooth cover by a connected scheme?
An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there ...
- 1,004
2
votes
0
answers
117
views
Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)
Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...
- 573
15
votes
0
answers
402
views
Grothendieck dessins d'enfants - current surveys or text you can recommend?
I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
1
vote
1
answer
168
views
Polynomial constraints on the values of continuous functions $\mathbb{R}\to\mathbb{R}$
Denote by $S$ the set of closed points in $X=\mathrm{Spec}\:\mathbb{R}[x_\alpha]$ ($\alpha \in \mathbb{Q}$) that have $\mathbb{R}$ as their residue field. There is an injective map from the set of ...
user158636
15
votes
5
answers
2k
views
Striking existence theorems with mild conditions, and simple to state: more recent examples?
I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
6
votes
0
answers
159
views
Whitney stratification for proper morphisms
Let $f: X \to \Delta$ be a flat, projective morphism, smooth over the punctured disc $\Delta^*:=\Delta \backslash \{0\}$ and central fiber $f^{-1}(0)$ is a reduced, simple normal crossings divisor. ...
- 1,583
6
votes
1
answer
450
views
Prove category of constructible sheaves is abelian
Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology. I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of ...
- 1,651
36
votes
1
answer
3k
views
Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
- 59k
1
vote
1
answer
153
views
Collineations of projective spaces and isomorphisms of fields
For a (topological) field $F$ by $FP^2$ we denote the projective plane, i.e., the quotient space of $F^3\setminus\{0\}^3$ by the equivalence relation $\vec x\sim\vec y$ iff $\vec x=\lambda\vec y$ for ...
- 40.2k
9
votes
3
answers
1k
views
Link of a singularity
I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$.
If we set $x = x_1+ix_2, y = y_1+iy_2, z ...
user56259
15
votes
2
answers
2k
views
Which definition of "proper" is better?
It is well known that topology and algebraic geometry assign different meanings to the word "proper".
Let us recall the relevant definitions from topology (and we work in the context of topological ...
- 18.1k
4
votes
0
answers
477
views
A slightly canonical way to associate a scheme to a Noetherian spectral space
Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
user139951
6
votes
2
answers
381
views
Do codimension 1 subsets of a scheme cover it?
Let $X$ be an irreducible scheme. A point $p\in X$ is said to have codimension $n\in\mathbb{Z}_{\geq 0}\cup \{\infty\}$ if $\overline{\{p\}}$ has codimension $n$. Is it true that any point of positive ...
user139794
9
votes
1
answer
384
views
The (co)tangent sheaf of a topological space
Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O _X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ ...
- 22.4k
0
votes
0
answers
57
views
A retract algebraic subset of the plane which does not admit an algebraic retraction
What is an example of an algebraic (=Zariski closed) subset $C$ of $\mathbb{R}^2$ which is a topological retract of $\mathbb{R}^2$, but there is no algebraic retraction $P:\mathbb{R}^2 \to C$?
What ...
- 280
6
votes
0
answers
345
views
Topological Singularities in Affine Varieties
Let $X$ be an affine variety over $\mathbb{C}$. Let $x\in X$.
If $x$ is non-singular, then $x$ is locally holomorphic (in the Euclidean topology). See here for a relevant MO post.
By results of ...
- 8,334
4
votes
1
answer
345
views
What is the topological/smooth analogue of Nagata compactification
A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a ...
- 10.3k
10
votes
1
answer
555
views
Noetherian spectral space comes from noetherian ring?
Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $\textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ...
- 2,863
1
vote
0
answers
116
views
Explicit description of the scheme obtained by relative gluing data over a base scheme
I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
- 403
8
votes
1
answer
513
views
Question about taking the Zariski closure in $\mathbb{A}_{\mathbb{R}}^n$
Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials.
Suppose $V \subseteq \mathbb{...
- 3,527
1
vote
0
answers
187
views
Clopen subsets of a closed subspace of a spectral space
Let $X$ be a topological space. Set
$K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called spectral,
if it satisfies all of the following conditions:
1) $...
- 11
1
vote
3
answers
988
views
Classical point-set topology using Grothendieck topologies
Its well known that the category of opens $O(X) $of a topological space $X$ can be endowed with a Grothendieck topology making it into a site. I am looking for references which take the reader through ...
1
vote
0
answers
109
views
Question regarding the image of a polynomial map containing a small box
I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously.
Let $\delta, \varepsilon > 0$.
Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
- 3,527