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
Questions tagged [projective-space]
1662 questions
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Lines in projective space
I have the following definitions:
Given a vector space $V$ over a field $k$, we can define the projective space $\mathbb P V = (V \backslash \{0\}) / \sim $ where $\sim$ identifies all points that lie on the same line through the origin.
A…
Jonathan
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Questions about the projective plane
On Wikipedia it is stated that
points of the form $[x:y:1]$ are the usual real plane and
points of the form $[x:y:0] $ are the line at infinity.
But this choice $z=0$ seems arbitrary to me. The projective space seems pretty symmetric so one might…
self-learner
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Definition of projective space
I am reading Silverman's 'The Arithmetic of Elliptic Curves'.
It says $\mathbb{P}^n(K)=\{[x_0,\cdots,x_n]\in\mathbb{P}^n:\text{ all }x_i\in K\}$, and then a remark says: if $P=[x_0,\cdots,x_n]\in\mathbb{P}^n(K)$, it does not follow that
each…
Eulerian
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The space of rays in R^n coming out of the origin
I have a quick question. Doing some research in Natural Language Processing I am coming across the space of rays in $\mathbb{R}^n$ coming out of the origin, in other words, the space $\mathbb{R}^n/\sim$ where $\sim$ is the equivalence relation…
gmauricio
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What is the right way to think of a line in projective space?
Let $k$ be an algebraically closed field. From calculus, I learned to think of the line through two points $p,q \in k^n$ as the span of $q-p$, plus a displacement vector (either $p$ or $q$).
$$
L_{pq} = \{ p + \lambda(q-p) : \lambda \in k\}
$$
I'm…
Joshua Ruiter
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Why $\mathbb{CP}^1 \cong S^2$?
Consider $\mathbb{CP}^1 =\frac{\mathbb{C}^2\setminus \{ 0\}}{\mathbb{C}^*}$, one dimensional projective Hilbert space. I was wondering if someone could help me about proving $\mathbb{CP}^1 \cong S^2$
Here $\mathbb{C}^*$ acts on …
Mahtab
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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 $[x:y:z]=[kx:ky:kz]$. I didn't understand what they…
green_blue
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About the $PG(3,2)$ square representation
I'm struggling since a while about understanding Curtis' MOG. To land there, I'm going through Cullinane's seminal square representation of $PG(3,2)$, as depicted here:
http://finitegeometry.org/sc/pg/dt/ortholatin.html
It's not clear to me how…
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What is the mapping from $\Bbb R^n$ to the (real) projective space?
I'm particularly interested in the case where $n=2$. The definition of $\mathbb RP^n$ as a manifold maps the point $(x_1,\ldots,x_n)$ in $\mathbb{R}^n$ by means of the function $\frac{x_i}{x_j}$ (where $i\ne j$) though it isn't specific about the…
Mr X
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How to show that $\mathbb{P}^n$ is a variety?
I would like to show that $\mathbb{P}^n$ is a variety, given that it is a prevariety. This is how I've started...
To show that $\mathbb{P}^n$ is in fact a variety, we must show that the diagonal of $\mathbb{P}^n$,$\Delta(\mathbb{P}^n) = \{(x,x): x…
sc636
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topological types of conics in real projective space of dimension 2
I know that all the conics have the same topological type in ${\mathbb{P}^2}_{\mathbb{R}}$. But I can not see why. Someone help me please.
for example, how do hyperbola and circle have the same topological type?
Kummer
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Projections from $\mathbb{C}^{n+1}$ to $\mathbb{CP}^n$. (check my reasoning)
Let $V,U$ vector subspaces of $\mathbb{C}^{n+1}$ of dimension $r+1$, $s+1$.
Let $\pi : \mathbb{C}^{n+1}/\{0\} \longrightarrow \mathbb{CP}^n$ the projection map to complex projective space defined by $\pi:(x_0, ... ,x_n)\mapsto(x_0:...:x_n)$.
This is…
AnonymousCoward
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Coordinate system in a Projective Space
How many coordinates does a point in $\mathbb{R}P^{n}$ have and which are the Homogenous Coordinates in $\mathbb{R}P^{n}$ ? To explain my question better. If we have a point p in the manifold $\mathbb{R}^{n+1}\smallsetminus\{0\}$ then we write p ,in…
Mathitis
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Visualisation of $\mathbb{P}^1(\mathbb{Z}/6\mathbb{Z})$
For some exercise I required the fact that there are $p + 1$ lines in $\mathbb{F}_p \times \mathbb{F}_p$, where a line is defined as a $1$-dimensional subspace. This is easy to see, if we draw $\mathbb{F}_p \times \mathbb{F}_p$.
However, I also want…
Krijn
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Calculating the dimension of a projective variety?
I am having difficulty determining the dimension of a projective variety in general.
For example, I am confused about the dimension of the projective variety $X-Y=0$ in $\mathbb{P}^3$.
I was thinking that the dimension of this variety is $1$…
