Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [projective-space]

Questions about projective space in geometry, a space which can be seen as the set of lines through the origin in some vector space. As such it is a special case of a Grassmanian. See https://en.wikipedia.org/wiki/Projective_space

0
votes
1answer
24 views

An injection yields an isomorphism in cohomology? $\Bbb RP^n \rightarrow \Bbb RP^{\infty}$.

In page 79 line 10 Hatcher made the claim that An embedding $\Bbb RP^{n-1} \rightarrow \Bbb RP^\infty$ induces an isomoprhism $$H^i(\Bbb RP^n, \Bbb Z_2) \leftarrow H^i(\Bbb RP^\infty, \Bbb Z_2)...
2
votes
1answer
59 views

How does the high school approach to classifying linear systems generalize to more variables, and why does it work?

If we have two linear equations in two variables, there may be $0$, $1$ or infinitely-many solutions. The standard approach to working out which case we're dealing with is probably to use Gaussian ...
0
votes
0answers
33 views

Spaces of matrices and projective spaces

Consider $\mathbb{C}^8$ i.e. 8 copies of the field of the complex numbers. I want to identify it with the set of matrices 2x4 $(z_{00},z_{01},z_{02},z_{03},z_{10},z_{11},z_{12},z_{13})$ now i have to ...
0
votes
1answer
16 views

Relationship between Affine definition of singular point and projective definition

Let $C : F(X,Y,Z) = 0$ be a projective curve given by a homogeneous polynomial $F \in \mathbb{C}[X,Y,Z]$, and let $P \in \mathbb{P}^2$ be a point. Prove that $P$ is a singular point of $C$ if ...
1
vote
1answer
36 views

A question about morphism of projective spaces

Consider the morphism: $$ f: (\mathbb{P}^2 -\{(0:0:1),(0:1:0) \} )\to \mathbb{P}^3 $$ Given by $f((x:y:z))=(x^2:xy:xz:yz)$, my problem is to find the closure of the image of $f$, my argument was: ...
0
votes
0answers
16 views

About the image of an affine variety into a projective space.

Assume to have a morphism $f: \mathbb{A}^n \to \mathbb{P}^m$, I want to compute the ideal of the the projective closure of $ f(\mathbb{A}^n)$. I think that in general the image of an affine variety ...
0
votes
0answers
18 views

What's the deal with projectively equivalent transformations?

Suppose that I have a homogeneous function from $\mathbb R^n$ to $\mathbb R$, where $ f(\alpha \mathbf{x}) = f(\mathbf{x}) $ for every scalar $\alpha$ and every vector $\mathbf x$ in $\mathbb R^n$. ...
1
vote
0answers
46 views

About a strategy to check smoothness of an algebraic variety

I want to ask about the Jacobi criterion for checking smoothness of this projective variety (i'll write the coordinates as $x,y,z,w$), I need to find singular points of: $$ xyz + xyw +xzw+ yzw=0 $...
2
votes
0answers
29 views

Dimension image of morphism of projective varieties

Let $f: \mathbb{P}^n \to \mathbb{P}^m$ be a rational map. Then there exists $U \subset \mathbb{P}^n$ open so that $f_{|U}$ is a morphism. What can we say about the dimension of $\overline{f(U)}$? We ...
0
votes
0answers
19 views

Find the dimension of an (explicit) projective variety

I have to compute the dimension of : $$ X=V(xz-y^2,xz-xw,yz-yw,z^2-zw) \subset \mathbb{P}^3(\mathbb{C}) $$ My idea was to compute irreducible components of this algebraic sets and intersect with some ...
0
votes
0answers
31 views

One-point compactification and algebraic geometry

An affine curve $C:\{(x,y)\in\mathbb{C}^2:Q(x,y)=0\}$ can always be extended to a projective curve $\tilde{C}:\{[x:y:z]\in\mathbb{CP}^2:P(x,y,z)\}$ where $P(X,Y,Z)$ is a homogenisation of the ...
0
votes
0answers
14 views

Cell structure of connected sum of complex projective planes

I am trying to find a cell structure for the connected sum $\mathbb{C}P^2\# \mathbb{C}P^2$. My approach is the following: Find a cellular structure for $\mathbb{C}P^2- \text{int}(D^4)$ where $D^4$ ...
0
votes
0answers
14 views

Unique projective maps preserving lines

Let $\mathbb{P}(V)$ be a projective space of dimension $3$ and let $L_i$, $i=1,2,3$ be pairwise non-intersecting lines in $\mathbb{P}(V)$. If $\phi: L_1 \to L_1$ is a projective transformation, prove ...
0
votes
0answers
23 views

find a vector with almost equal projection onto multiple vector

I have reached to a problem and I appreciate any suggestion to solve this issue. I need to find a vector like $v\in \mathbb{C}^M$ which has almost same projection onto multiple vectors with the same ...
1
vote
1answer
40 views

Show that this linear projection with center is an isomorphism

Let $n \in \mathbb{N}$ and $f,g \in k[x_1,x_2]$ homogenous and coprime polynomials where $\deg (f)=n$ and $\deg(g)=n-1$. Let $C$ be the variety $V(f-x_0g) \subset \mathbb{P}^2$. Show that the linear ...
0
votes
0answers
26 views

How do I get a plane from two perpendicular cubes in 4d space?

I am trying to get a notion of 4D space. Since perpendicular intersections of 1D Lines in a Plane point to a Point and perpendicular intersections of 2D Planes in a 3D Space point to a Line, I figured ...
1
vote
0answers
33 views

Ordinary r-fold point on the dual curve

Let $C$ a projective curve in $\mathbb{P}^2(\mathbb{C})$. We say a line $L \subset \mathbb{P}^2$ is mulltiple tangent of $C$ if there are $P_1, \dots P_k$ points on $C$ such that $L$ is the tangent of ...
5
votes
1answer
87 views

$\mathbb{C}P^1\times…\times \mathbb{C}P^1/S_m=\mathbb{C}P^m$

Let $X:=\mathbb{C}P^1\times...\times \mathbb{C}P^1$ be the product of $m$ copies of $\mathbb{C}P^1$ and $S_m$ acts on $X$ by permuting the factors. Then why is $X/S_m=\mathbb{C}P^m$?
1
vote
0answers
30 views

How to find the equation of an image under a central projection

Let $\pi:\mathbb{P}^3 \to V(x_2) \cong \mathbb{P}^2$ the linear projection with center $P =(0:1:0:0)$. Find the equation for the image of $C=\{(s^3:s^2t:st^2:t^3)|~(s:t) \in \mathbb{P}^1 \}$ under $\...
3
votes
1answer
33 views

Affine charts are dense in projective space

Given a field $k$, we define the scheme-theoretic $n$-th affine space over $k$ by $\mathbb{A}^n_k=\text{Spec}(k[X_1,\dots,X_n])$ and the $n$-th projective space over $k$ by $\mathbb{P}^n_k=\text{Proj}(...
0
votes
1answer
47 views

Equivalent Definitions of Lines in Projective Space

I’ve been working with two definitions of lines in $\mathbb{P}_\mathbb{R}^2$, and tried to show their equivalence. The first is that, given two points $a=(a_0:a_1:a_2)$ and $b=(b_0:b_1:b_2)$, the ...
3
votes
1answer
140 views

Is every map from $\mathbb{R}P^2 \to S^2$ nullhomotopic?

I think the answer is no, but I'm not sure. Consider the CW-structure on $\mathbb{R}P^2$ given by one $0$-cell $x$, one oriented $1$-cell $a$ attached to $x$ at both ends, and one $2$-cell whose ...
1
vote
0answers
49 views

Easy question about computing points of infinity

Let us consider the curve $C$ determined by $x^2t^2 + y^2t^2 + x^2y^2 - t^4$ over $F$ ( finite field with $|F|=q$ ). Maybe it is a stupid question but how can I compute the points at infinity? The ...
0
votes
0answers
20 views

The contact structure on the real projective space

I came across the following description of the standard contact structure on the real projective space: let $\mathbb{R}^{2n}$ be endowed with its standard symplectic structure. The real ...
1
vote
1answer
29 views

What transformations can be set by projecting a straight line onto a straight line

What transformations can be set by projecting a straight line onto a straight line (without adding an infinitely distant point)? I said that the homothety with coefficient $k \neq 1$ and the ...
1
vote
1answer
60 views

Pushforwad bundle alond a degree 2 map from $P^1$ to $P^1$

Suppose $f:P^1\rightarrow P^1$ is a degree 2 morphism. Let $L$ be a line bundle on $P^1$ (which is equivalent to a $O(n)$), then what is $f_*L$? As for an open set $U$ we can see the preimage is a ...
4
votes
0answers
75 views

Computing de Rham cohomology group $H^1(\mathbb{RP}^2\#\mathbb{RP}^2)$

I am trying to compute the de Rham cohomology group $H^p(\mathbb{RP}^{n+1}\#\mathbb{RP}^{n+1})$ and I am stuck at computing $H^1(\mathbb{RP}^2\#\mathbb{RP}^2)$. ($\#$ stand for the connected sum) ...
1
vote
1answer
31 views

Coordinate representation of tangent vector to the real projective plane.

Let $a, b, c$ be three real numbers. Let $X=\gamma'(0)$ be a field on the projective plane $P^2 (\mathbb{R})$, which has the usual maps: $$ \phi_1 ([1 : x : y]) = (x, y), \qquad \phi_2 ([x :...
0
votes
0answers
11 views

Ratio of “proportional to” equations equaling each other

I'm reading this paper and came across this section, which I don't completely understand. Equations (19) and (20) are "proportional to" equations, where $LHS = k RHS$. Equation (22) is formed using ...
0
votes
1answer
22 views

Curves and divisors in weighted projective planes

Let us consider the weighted projective plane $\mathbb{P}(q_0,q_1,q_2)=\mathrm{Proj}(\mathbb{C}[x_0,x_1,x_2])$ where $x_i$ has weight $q_i$ for every $i\in \{0,1,2\}$. Let $f\in \mathbb{C}[x_0,x_1,x_2]...
1
vote
0answers
35 views

Tensor product of sheaves over weighted projective spaces

I have read that if we have a weighted projective space $\mathbb{P}$ in general it is not true that: $$ \mathcal{O}_\mathbb{P}(n)\otimes \mathcal{O}_\mathbb{P}(m)\simeq \mathcal{O}_\mathbb{P}(n+m). $$ ...
0
votes
0answers
35 views

Computing the induced map on homology from projective space

Define a map from $F:P^1(\Bbb C)\times P^1(\Bbb C) \rightarrow P^2(\Bbb C)$ by $((x,y), (z,w)) \mapsto (xz, xw + yz, yw)$ What is the induced map on homologies? $$H_p(\Bbb{CP}^1 \times \Bbb{CP}^1) \...
4
votes
4answers
351 views

Lie group structure on the complex projective space

There is a famous theorem about when $S^n$ has the structure of a Lie group. What about the complex projective space $\mathbb CP^n$? For example, why $\mathbb CP^2$ is not a Lie group (without using ...
1
vote
0answers
101 views

Identifying the Plane at Infinity in the World Necessitates Determining the Affine Geometry of the World?

Page 18 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: 1.8 Auto Calibration $\vdots$ Generally ...
0
votes
0answers
9 views

Infinitely many fixed-rank bundles on projective space?

I was asked as a question whether a topological space can have infinitely many non-isomorphic bundles (with fixed rank). As a hint we were recommended to consider projective space. I don't really know ...
1
vote
0answers
12 views

Plot projective curves

Do you know any website, app or program that can plot curves in the projective plane? I know the projective plane $\mathbb{P}^2 \mathbb{R}$ can be visualized as a sphere with the antipodal points ...
2
votes
0answers
41 views

Find a projective change of coordinates that maps the (projective) line L to the (projective) line G

Let $L$, $G \subset \mathbb{P}^2$ be lines. Show that there exists a projective change of coordinates $T$, such that $T(L)=G$. This is how we defined a projective change of coordinates in $\mathbb{P}...
1
vote
1answer
45 views

Computing Fubini-Study metric from the formal definition

Definition: the Fubini-Study metric $g_{FB}$ on $\mathbb{CP}^n$ is the only metric which makes the projection $\pi:(\mathbb{S}^{2n+1},g)\to(\mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is ...
0
votes
1answer
37 views

Submanifold of real projective space

Would you like to tell me how to prove that $\{[x_{0}:x_{1}:x_{2}]: x_{0}x_{1} + x_{2}^{2} = 0 \} \subset \mathbb{R}\mathbb{P}^{2}$ is a submanifold (of $\mathbb{R}\mathbb{P}^{2}$)?
1
vote
0answers
13 views

Example for intersection, multiplicity and tangents for projective plane curves

We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, ...
1
vote
0answers
36 views

Non split real form of projective space

On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this ...
2
votes
1answer
47 views

“line at infinity” in projective plane

("Algebraic Geometry: A Problem Solving Approach" by Thomas Garrity) I am struggling with Exercise 1.4.12.1 in the above, which I quote with some context: Here is my intuitive thinking: (a) lines ...
0
votes
1answer
35 views

Cohomology of Projective Space $\mathbb{PR}^n$ with Coefficients in $\mathbb{Z}/2$

We know the cohomology ring with coefficients in $\mathbb{Z}/2$ of projective real space $\mathbb{PR}^n$ is $$H^*(\mathbb{PR}^n, \mathbb{Z}/2) = \mathbb{Z}/2[X]/(X^{n+1})$$ with graduated generator ...
1
vote
0answers
36 views

Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies ...
0
votes
0answers
29 views

Proving that a point is the result of only two lines intersecting and a line is the result of only two points being aligned

Let $S_0$ be a set of four points in the real projective plane such that any three points of $S_0$ are not aligned. Let $L_0 := \emptyset$. For every integer $n \ge 1$, we define the following: ...
0
votes
0answers
11 views

Projective subspace of same dimension

I'm looking at families of curves in $\mathbb{P}^2$ (over $\mathbb{C}$), specifically the set $\mathcal{L}_d$ of projective curves defined by a homogeneous polynomial $P \in \mathbb{C}[x_0,x_1,x_2]$ ...
1
vote
1answer
32 views

Projective/ Finite Geometric Basics!

I'm taking intro to coding theory and am having some trouble understanding the basics of Projective Geometry, since our text does not give it much discussion. Namely, if PG(r-1,q) is the set of all ...
2
votes
0answers
28 views

How to check for a projective space?

When I have a division ring commutative its pretty straight forward! But when its not commutative then I'm stuck, it's possible that in this case its not a projective plane? Can someone give a ...
1
vote
1answer
42 views

Proving Hopf's Fibration $\pi: \mathbb{S}^{2n+1}\to\mathbb{CP}^n$ is a submersion

Prove that the following map is a smooth, surjective submersion: \begin{align*} \pi:\mathbb{S}^{2n+1}&\to\mathbb{CP}^n\\ (x_0,y_0,...,x_n,y_n)&\mapsto [x_0+iy_0:...:x_n+iy_n] \end{align*} ...
4
votes
2answers
257 views

If All the Points Lie On a Plane, Then Why Does the Linear Mapping Reduce to …?

I previously asked a question with regards to what the matrix $\mathrm{H}_{3 \times 3}$ is/represents in the following textbook excerpt: In applying projective geometry to the imaging process, it ...