3
votes
1answer
45 views

Are any (non-empty) Euclidean open sets dense in the Zariski topology?

It's well known and easy to show that every Zariski open set is dense in the Zariski topology. However I search the web and didn't find an answer to my question, which I believe is true. My ...
0
votes
0answers
16 views

The normalization of a product of varieties

Let $X,Y$ reduced complex analytic spaces, $X^{'}$ and $Y^{'}$ the normalizations of $X$ and $Y$, respectively. Let $(X \times Y)^{'}$ the normalization of $X \times Y$. Is true that $(X \times Y)^{'} ...
3
votes
1answer
55 views

When do functions turn a space into a locally ringed space?

Let $X$ be a topological space, and consider for each open set $U \subseteq X$ a set $F_U$ of functions $U \to k$ into some fixed field $k$. Let $\mathcal{O}$ be the sheaf of $k$-algebras induced by ...
3
votes
1answer
54 views

Exercise 3.20 in Hartshorne on dimension of integral schemes of finite type over a field.

While working on exercise 3.20 in Hartshorne I've gotten stuck on something for a while. Given an integral scheme $X$ of finite type over a field I'm tring to show the existence of some affine open ...
1
vote
1answer
53 views

Irreducible components of fiber bundle

Suppose $\pi:X \rightarrow Y$ is a (locally trivial) fiber bundle $F$, where all spaces are Noetherian. Suppose $F$ and $Y$ are irreducible; show that $X$ is also irreducible. Here a similar ...
4
votes
0answers
62 views

Openness of a subset in complex 2-plane

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is ...
4
votes
1answer
46 views

If $G/H$ and $G$ are connected linear algebraic groups must $H$ also be connected?

Let $k$ be a perfect field (e.g of characteristic zero) and let $G$ and $H$ be linear algebraic groups over $k$, with $H$ a normal subgroup of $G$. If both $G$ and $G/H$ are connected, must $H$ ...
4
votes
0answers
129 views

Standard spaces with non-standard topology and around

The following series of questions gives me no rest. Let $\mathbb{\widetilde{R}}^n$ be $\mathbb{R}^n$ with Zariski topology, i.e. we say that $A\subset \mathbb{R}^n$ is closed if it is given by the ...
0
votes
0answers
49 views

Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
3
votes
0answers
110 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
2
votes
1answer
105 views

The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical ...
2
votes
0answers
43 views

explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
0
votes
1answer
38 views

an equivalent statement of “morphisms of projective varieties are closed”

I am interested in seeing why the statement (1) "If $Y$ is any variety and $Z$ a closed subset of $\mathbb{P}^n \times Y$, then the projection of $Z$ on $Y$ is closed." implies the statement ...
1
vote
1answer
77 views

Constructible sets

Is it possible to write down all the constructible sets in $\mathbf{C}$ (endowed with the Zariski topology) or some other "simple" space?
0
votes
0answers
25 views

The closure of semialgebraic sets is semialgebraic.

I want to prove that the closure of semialgebraic subsets of $\mathbb{R}^n$ with respect to the Euclidean topology is semialgebraic. I may use the Tarski–Seidenberg theorem. Please give me not the ...
1
vote
1answer
22 views

Is this subset of $PSL(n,\mathbb{R})$ Zariski-closed?

For some non-identity element $[A]\in PSL(n,\mathbb{R})$ ($[A]$ being the class of $A\in SL(n,\mathbb{R})$) and linearly independent vectors $x,y\in\mathbb{R}^n$, let $[x],[y]$ denote the classes of ...
2
votes
0answers
31 views

How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
1
vote
0answers
64 views

Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
9
votes
1answer
117 views

Topological Dimension via chains of connected nowhere dense closed sets

In algebraic geometry, one defines the dimension at a point of a variety $X$ as the length of the longest chain of irreducible closed subsets (in the Zariski topology of the variety) containing the ...
0
votes
0answers
52 views

Homeomorphisms on Zariski topologies

I'm looking for a continuous bijection from a compact space to a non-Hausdorff topological space which isn't a homeomorphism. Since the identity $f:\mathbb{Z}\rightarrow\mathbb{Z},\ x\rightarrow x$ is ...
3
votes
1answer
49 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
3
votes
1answer
37 views

Image of Regular Map

Determine the image of the regular map $f: A^2 \to A^2$, $f(x,y)=(x,xy)$ and describe it from the point of view of topology. Would the image of f be $A^2$, because every point of $A^2$ is still in the ...
1
vote
2answers
100 views

Product topology of Affine Varieties

I want to prove that $X \times Y \subseteq \Bbb A^{2n}$ is an affine variety, given that $X,Y \subseteq \Bbb A^n$ are affine varieties. Is this proof correct? Since both $X$ and $Y$ are affine ...
2
votes
1answer
50 views

Open subset in Irreducible Topological Space is dense.

Show that every non-empty open subset of an irreducible topological space is dense. I know a lemma that states that $U \subset$X is dense iff for all $A \in \tau$, $A \cap U \neq \emptyset$. So ...
1
vote
2answers
128 views

Closed sets in Zariski Topology

What are the closed sets on $\mathbb{Z}$, $\mathbb{R}[x]$, $\mathbb{Z}[x]$? I was given this question, but it seems trivial, because aren't the closed sets only the affine varieties? So for each is ...
2
votes
1answer
29 views

Density, Irreducible Topological Space

Let X is a irreducible noetherian topological space, and $U \subseteq X$ is a nonempty open subset. If $B \subseteq U$ is dense, then so is $B \subseteq X$. Is this true or false, and why? Here ...
1
vote
1answer
53 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
2
votes
1answer
67 views

What topology has $Pic(X)$?

Let $Pic(X):=\{\mbox{Holomorphic line bundles on } X\}/\sim$ be the group of isomorphism classes of line bundles on $X$. It is well-know and easy to prove that, working in Cech cohomology, $Pic(X) ...
1
vote
0answers
33 views

dimension of a subspace of a flag variety

Let $X$ be a topological space. If $X = \bigcup U_\alpha$ is an open covering of $X$ then $$\dim X = \sup_\alpha \dim U_\alpha.$$ Now suppose that $X = \coprod U_\alpha$, i.e., $X$ is the disjoint ...
0
votes
1answer
50 views

The torus as a complex variety

I'm interested in the topological torus, ie. the homeomorphism class of $S^1\times S^1$. Clearly, it can be realized as the real algebraic variety in $\mathbb{R}^4$ as the solution set to ...
0
votes
1answer
98 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
6
votes
3answers
157 views

Are the fibers of a flat map homotopy equivalent?

At the end of the Wikipedia article on Deformation Retract, there is the following sentence: Two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger ...
7
votes
1answer
98 views

Any Real Algebraic Variety Has Finitely Many Path Components?

Consider the following statement: Statement: Any real algebraic variety has only finitely many path components. By 'path component' I mean: Let $X$ be a real algebraic variety. Then $X$ is a subset ...
4
votes
1answer
57 views

Homeomorphism class of GL_n?

For example, it is easy to see that $GL_1(\mathbb{C})$ is a plane minus a point, and $GL_2(\mathbb{R})$ is $\mathbb{R}^4$ with a topological (half-open) cube removed (since the matrices of determinant ...
2
votes
2answers
84 views

Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
0
votes
1answer
45 views

intersection of two irreducible spaces not irreducible

Let $X$ be a topological space and let $Y_1,Y_2$ be two distinct irreducible subsets with none containing the other. Then it seems to me that $Y_1 \cap Y_2$ need not be irreducible. Is that right? ...
2
votes
1answer
91 views

Zariski topology and polynomial maps

I've read on my book that Zariski topology is coarser than every topology in which polynomial maps are continous, but no proof of this fact is given. Could someone sketch me the proof of this?
3
votes
2answers
43 views

The two projection maps are different?

I'm reading Vistoli's notes, and I came across something on the bottom of page 31, starting with "The reader might find our definition of sheaf pedantic..." Essentially my problem is the following ...
3
votes
1answer
145 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
2
votes
1answer
82 views

$X$ being locally closed is not equivalent to every extension by $0$ of a sheaf $F$ being unique?

In Tennison's book "Sheaf Theory", the author presents a proof that there is a unique extension by $0$ for a sheaf $F$ over $X$ iff $X \subset Y$ is locally closed . However, apparently in the proof ...
5
votes
1answer
106 views

What Information/Advantage do we Gain by Substituting a Continuous Map by a Fibration?

I'm trying to understand the usefulness of "substituting" a continuous map f , by a fibration F. By substituting, I mean there is the result that given a continuous map $f:X \rightarrow Y $ , for ...
1
vote
0answers
45 views

Let $R$ be a conmutavive ring. Prove the following: If $Spec(R)$ is $T_1$ then $Spec(R)$ is Hausdorff.

Let $R$ be a conmutavive ring. Prove the following: If $Spec(R)$ is $T_1$ then $Spec(R)$ is Hausdorff. Here $Spec(R)$ means the Zariski topology over the set of all prime ideals of $R$. To put it ...
0
votes
1answer
43 views

Irreducibility of set

Let $Y$ be a subset of the topological space $X$ and let $\{U_i\}$ be an open cover of $X$. 1.If $Y$ is not contained in any $U_i$ then $Y$ is irreducible? 2.If $\{V_j\}$ is an open cover of $Y$ ...
0
votes
1answer
32 views

Support of form and embedded varieties

I need help with some inclusions. Let $i: S \rightarrow M$ be an embedding between two oriented varieties of dimension k and n respectively. Assume that the $i(S)$ is closed and that $\omega\in ...
-1
votes
1answer
55 views

Irreducible space with infinitely many irreducible components

It would be intuitively satisfying to say the following: A topological space is irreducible if and only if it has exactly one irreducible component. But it is not immediately clear how to prove ...
2
votes
1answer
56 views

Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
1
vote
1answer
102 views

If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen.

If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen. This is exercise 3.6P of Vakil. I can see that a union of connected components is closed. This is ...
4
votes
1answer
47 views

Noetherian topological subspaces

I'm trying to prove that any subset of a noetherian topological space is noetherian in its induced topology. MY ATTEMPT OF SOLUTION Let $X$ be a topological space and $Y$ a subspace of $X$. If ...
2
votes
1answer
29 views

About closed map beween schemes

Probabily it's trivial but I've no idea for a proof. Let $f: X \rightarrow Y $ a continuous map between Topological Spaces, with $Im(f)$ closed in $Y$. I know there exist a covering $\{Y_i\}$ of $Y$ ...
1
vote
1answer
76 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...