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

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25 views

Homology of $\mathbb{R}P^n \times \mathbb{R}P^m$

If we know the homology of $\mathbb{R}P^n$, how can we determine the homology of $\mathbb{R}P^n \times \mathbb{R}P^m$ (where $n$ and $m$ are natural numbers)?
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Singularity of the curves in affine form vs in projective form

I have just started learning about elliptic curves and I have this thought about curves in affine form and projective form. Apologies in advance if the question sounds silly, admittedly I am not ...
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Are all isomorphisms between the Fano plane and its dual of order two?

Famously from every finite projective plane $P$ we can create a dual projective plane $P'$ by taking the points of $P'$ to be the lines of $P$ and the lines of $P'$ to be the points of $P$. It is also ...
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Taking the boundary of a tubular neighborhood

By $\mathrm{Tub}({\bf RP}^2)$, I mean a small tubular neighborhood of the standardly-embedded ${\bf RP}^2$ in ${\bf CP}^2$ (embedded as the fix-point set of conjugation $\mathrm{conj}:{\bf CP}^2\to{\...
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Coordinates in projective space

I have just started learning projective geometry. There they define a projective point to be a line passing through origin alternatively they added that it can be said to be $[x:y:z]$ satisfying $[x:y:...
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A Morse function on $\mathbb RP^2$ whose all critical points are non-degenerate. [duplicate]

$\mathbf {The \ Problem \ is}:$ Find a Morse function on $\mathbb RP^2$ with one non-degenerate critical point . It may have other critical points . $\mathbf {My \ approach}:$ If we take a homogenous ...
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Projective Geometry interpretation of a set

I have the following example: Lets see the projective straight line $l \subset \mathbb{P^2}$ that goes through the points $[1:1:0]$ and $[1:0:1]$ (first is this any diferent that $(1,1,0)$ and $(1,0,1)...
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Signed Morse homology of $\mathbb{R}P^2$

Here's my understanding of computing the signs in Morse homology (following the book by Audin and Damian). Let $f$ be a Morse function on a manifold $M$ with a negative pseudo-gradient $X$. Let $W^s(p)...
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Prove that $\mathbb R\mathrm{P}^n$ is orientable if and only if $n$ is odd

I am trying to prove that: The real projective space $\mathbb R\mathrm{P}^n$ is orientable if and only if $n$ is odd. For do so, consider first the antipode map $\sigma:\mathbb R^{n+1}\to \mathbb R^{...
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proving a result in (elementary) planar projective geometry about cross ratios

I'm stuck on the following problem in the projective space $P(\mathbb{R}^3) = \mathbb{R}P^2$. Say we have three distinct and non-concurrent lines $\mathcal{A}, \mathcal{B}$ and $\mathcal{C}$. Let $P$ ...
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Simplify $ch(\mathcal{O}(1))td(\mathbb{P}^n)$

Let $\mathcal{O}(1)\to \mathbb{P}^n$ be the the inverse of the tautological bundle over the projective space. Denote by $td(\mathbb{P}^n)$ the Todd class of the tangent bundle to $\mathbb{P}^n$. Can ...
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In projective space, map mapping a plane containing a line to its intersection with a disjoint line is a homography

If $\mathbb{P}(V)$ is a projective space of projective dimension $3$ and $D$, $D’$ are two disjoint projective lines in $\mathbb{P}(V)$, then we can consider the subset $D^*\subset \mathbb{P}(V^*)$ of ...
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How to visualise projective points and hyperplanes?

I am reading "Algebraic Geometry: A First Course" by Joe Harris and frankly I am very confused about projective $k$-planes, or maybe in general how points in projective space behave. I ...
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Calculating the pole of an hyperplane with respect to a projective quadric

In $\mathbb{P}(\mathbb{C}^4)$, I’d like to calculate the pole of the projective hyperplane $\textbf{H}$ defined by $X_2=0$ with respect to the quadric $\Gamma$ given by the equation $$X_0X_2+X_1X_2-...
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Finding a non-degenerated quadric containing three projective lines in $\mathbb{P}^3(\mathbb{C})$

I’m considering the disjoint sets in projective space $\mathbb{P}^3(\mathbb{C})$ defined as follows: $L_1$ given by the equations $X_1=0$ and $X_2=0$ $L_2$ given by the equations $X_0=X_1$ and $X_2=...
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No map $\mathbb{C}P^n \rightarrow \mathbb{C}P^m$ inducing a nontrivial map $H^2(\mathbb{C}P^m;\mathbb{Z}) \rightarrow H^2(\mathbb{C}P^n;\mathbb{Z})$

This is part of an exercise from Hatcher's Algebraic Topology text. The exercise is as follows: Using the cup product structure, show there is no map $\mathbb{R}P^n \rightarrow \mathbb{R}P^m$ ...
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Why does the projection $\pi:(a,b,c) \in S^2 \to[a,b,c] \in P^2$ have rank 2 everywhere?

Consider the differentiable function $g: [a,b,c] \in \mathbb{P}^2 \to (bc,ac,ab) \in \mathbb{R}^3$ It is not an immersion because it has rank 2 everywhere except in 6 points. To see this consider the ...
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What are the subgroups of the projective general linear group of degree 3 over a finite field?

I'm investigating something and the question of what the subgroups of $ \text{PGL}(3,\mathbb{F}_p)$ are has come up. I've found this Group Properties Wiki page, but it doesn't contain much useful ...
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The trace as an integral over the projective space

Let $(E,h)$ be a Hermitian vector space of dimension $n$ and $u\in End(E)$. We have an expression of the trace of $u$ as the integral $$Tr(u)=\frac{n}{A}\int_{S}\langle v,uv\rangle d\mu$$ where $S$ is ...
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How projective plane is used for projecting from the plane?

I asked this question to understand difference between projection plane and projective plane. But below the @bubba's answer, the user @Acccumulation comments this text which given below: The ...
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What is correct representation of homogenous coordinates?

I read from this answer and understand that the homogenous coordinates is of the form $[x, y, z]$ which represents by square bracket. Also read from this answer the homogenous coordinates is of the ...
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1answer
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Show compactification of positive real plane is homeomorphic to $\Bbb R \Bbb P^2$

$\Bbb R \Bbb P^2$ can be thought of as a compactification of $\Bbb R^2,$ and is formed by taking the quotient of $\Bbb R^{3}-\{0\}$ under the equivalence relation $x\sim \lambda x$ for all real ...
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Finding a ray plane intersection in projective space

I'm reading Proof 13.1 (page 344) in Hartley and Zisserman's Multiple View Geometry and I don't quite understand the following result. To compute $H$ we back-project a point $x$ in the first view and ...
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What can we get via taking quotient of $\mathbb{C}P^1$ by a finite abelian group?

Let $G_n = \langle\tau\rangle$ and $G_m = \langle\sigma\rangle$ be groups of $n$-th and $m$-th roots of unity. Define action of $G_n \times G_m$ on $\mathbb{C}P^1$ as follows. $$ (\tau^k, \sigma^t) \...
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Proving a generalization of Desargues’ theorem in dimension $\geq 2$

I need help with this generalization (I think) of Desargues’ theorem in higher dimensions: Let $E$ be a vector space of dimension $n \geq 2$ and let $D$, $D’$ and $D’’$ be three different projective ...
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1answer
38 views

Showing that an involution of the projective line with a fixed point fixes exactly two points

My idea to show this is to notice that a projective transformation is in general given by the form $\phi(z)=\frac{az+b}{cz+d}$, then $\phi(z)=z$ gives a second order equation with at most two ...
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333 views

What is topological distortion?

I understand the view confusion and struggling to understand topological distortion. I read from this website that the points of the plane that is parallel to the view plane and also passes through ...
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why does every line in projective complex plane pass through 2 inflection points of the curve C pass through a 3rd inflection point of the curve C

We are learning about cubic curves in the projective complex plane and saw this property and wasn't sure why it was true. Every line in the projective complex plane that passes through two inflection ...
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1answer
60 views

Definition of $ \ \mathcal{O}_{\mathbb{P}_k^n}(1)$ [duplicate]

I've found references to the object $\ \mathcal{O}_{\mathbb{P}_k^n}(1), \ $ where $\mathbb{P}_k^n \ $ is the usual scheme, but I'm not able to find its proper definition. Is it just a line bundle ...
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Intersection of a projective curve and a hyperplane

Let $\mathcal{C}$ be a projective curve on $\mathbb{P}^2$ over a field $k$ defined by $\Phi(x,y,z)$ such that it is homogenization of an affine curve $\Phi(x,y)$. When we look at the intersection of $\...
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Finitely vanishing vector field on the real projective plane

I am trying to find a vector field on \begin{align} \mathbb{P}_\mathbb{R}^2 := \mathbb{R}^3 \setminus \{(0,0,0)\} \big/ \sim, \end{align} where $\sim$ is the equivalence relation for $z, z'\in \mathbb{...
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1answer
64 views

Show that the dimention of the intersection of projective linear sub-spaces of dimentions $d_1$ and $d_2$ of $\mathbb{P}^n$ is bigger than $d_1+d_2-n$

Proposition: Let $L$ and $M$ linear projective subspaces of $\mathbb{P}^n$ of dimention $d_1$ and $d_2$ respectively. Prove that $\operatorname{dim}(L\cap M)\geq d_1+d_2-n$ (we consider the dimention ...
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1answer
101 views

Proving ${\mathbb{P}}^n$ is Hausdorff

I am trying to understand and complete the proof that the real projective space ${\mathbb{P}}^n$ is Hausdorff.In my notes it is modeled as${\mathbb{R}}^{n+1}\setminus \{0\}/\sim $ and it goes like ...
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1answer
100 views

What's the definition of an isomorphism between projective planes? [closed]

For some reason I can't seem to find a complete definition of what exactly it means for a map to be an isomoprism between two projective planes $f: P^n \to P^n$. I assume that f has to be bijective ...
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How many elements are in the projective line $\mathbb{P}^{1}(k)$ if k is a finite field

Assume k is a finite field with n elements, how many elements are in the projective line $\mathbb{P}^{1}(k)$ and how do I work this out? I know that an element of $\mathbb{P}^{1}(k)$ is represented by ...
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2answers
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Diffeomorphism between $\mathbf S^2 \times \mathbf S^2$ and complex projective hypersurface

I have a problem while studying differential geometry and I have a question. I want to show that $\{[{z_0:z_1:z_2:z_3]:z_0^2+z_1^2+z_2^2+z_3^2=0}\}$ and $\mathbf S^2 \times \mathbf S^2$ are ...
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Show the linear system of conics in $P\mathbb{K}^2$ tangent to a line L at P has codimension 2

My algebraic geometry class has just started on linear systems and while I follow most of the lemmas in our book (Elementary Geometry of Algebraic Curves by Gibson) on codimension and dimension of ...
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37 views

Show that there exists a function such that all points $P_i \in k^2$ are mapped to some $\lambda_i \in k$

To be precise, the question is : Let $k$ be a finite field. Let $P_{1}, \ldots, P_{m} \in k^{2}$ be distinct points and let $\lambda_1, \ldots \ldots, \lambda_{m} \in k$; show that there exists $f \in ...
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Plotting algebraic curves in ${\bf CP}^2$

I am considering a nwhere-singular degree $m$ polynomial $A\in{\bf R}[X,Y]$. I take its homogeneization to be ${}^h\!\!A(X,Y,Z)=Z^mA(X/Z,Y/Z)$. This means that I can look at the zero-locus of $A$ in a ...
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In projective geometry, how does the choice of a hyperplane at infinity affect the structure of the affine space it defines?

Let $E$ be a vector space, $H\subset E$ a vectorial hyperplane, and $\mathcal{H} \neq H$ an affine hyperplane parallel to $H$. Then I know that every vectorial line (one-dimensional subspace) $D \...
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32 views

projective submanifold

Given the smooth manifold $M = RP^2 \times \Gamma$ where $RP^2$ is the real projective plane and $\Gamma$ the unitary cylinder, verify that $$Z := \left\{ \big((y_0: y_1: y_2), (x_1, x_2, x_3)\big) \...
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1answer
62 views

What is the moduli space $M _{0,5}$?

Let $M_{0,n}$ denote the moduli space which consists of genus 0 non singular projective curves with n distinct marked points upto marked point isomorphism. So $M_{0,3}$ is a singleton set as any 3 ...
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1answer
54 views

Union of $\{[v^3]: v\in V\}$ and $\{[v^2w]:v,w\in V \text{ linearly independent}\}$ is a quartic surface in $\mathbb{P}^3$

I want to do Exercise 10.10 in J. Harris book "Algebraic Geometry", which is: "Let $V$ be a 2-dimensional vector space over a field $K$ with char$(K)\neq 2,3$, let $C=\{[v^3]: v\in V\}$,...
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What is the Zariski Closure of $C=\{[x:y:z] \in \Bbb P^2 \mid x=1,y=0\}$ in $\Bbb P^2$?

I want to determine the Zariski Closure of $C=\{[x:y:z] \in \Bbb P^2 \mid x=1,y=0\}$ in $\Bbb P^2$. My guess is that $\bar C=\{[x:y:z] \in \Bbb P^2 \mid y=0\}$, for this assume that $E=\{[x:y:z] \in \...
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1answer
34 views

Cartier Divisor for Hyperplane

Let $H$ be the hyperplane in $\mathbb P^n$ defined by $x_0=0$ What is the Cartier Divisor $(U_i,f_i)$ corresponding to H? I was thinking that the $U_i$ should be the distinguished open sets $D_+(x_i)...
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42 views

Gluing of $\operatorname{Spec} k[x,y], \operatorname{Spec} k[x^{-1},x^{-1}y],$ and $\operatorname{Spec} k[y^{-1},xy^{-1}]$

In Fulton's book on toric varieties he (on page 7) goes through the example of $\mathbb{P}^2$. He considers the fan given by $((1,0),(0,1),(-1,-1))$ and shows via its dual that the variety ...
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1answer
50 views

Projective subvarieties not contained at infinity

I have an oddly specific question that hopefully someone can help me with. For some context, let $\mathbb{A}^n$ be the $n$-dimensional affine space over an algebraically closed field, and $\mathbb{P}^...
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29 views

Find an explicit example of a plane curve of degree $4$ that has more than $3$ singular points.

I am taking an introductory Algebraic Geometry course and am considering the above problem. My understanding is that any reducible curve will have more than 3 singular points, so for this problem, ...
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92 views

Prove that the unique metric that makes the map from the complex plane with infinity the complex projective line an isometry has this formula:

Firstly let us define the map $\iota: \mathbb{C}_\infty \to P^1(\mathbb{C})$ by $\iota(z) = [z:1]$, $\;\iota(\infty) = [1:0]$. Here the equivalence class of $(z_1,z_2)$ is denoted $[z_1:z_2]$ where $(...
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1answer
47 views

Why ample line bundle is positive? (and recommendation about understanding the Kodaira embedding theorem)

I'm beginner of Complex Geometry. Please Understand. I'm learning the Daniel Huybrechts's Complex Geometry. I want to understand the Kodaira Embedding Theorem as soon as possible since it provoke ...

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