Articles of projective space

projective geometry hyperplane

For $j=0,\ldots,n$ consider the affine hyperplane $A_j:=e_j+\langle e_0,\ldots,e_{j-1},e_{j+1},\ldots,e_n\rangle$ in $\mathbb K^{n+1}$ and the associated embedding $\tau_j:\mathbb K^n\rightarrow\mathbb KP^n, \tau_j(x_1,\ldots,x_n):=[x_1:\ldots:x_j:1:x_{j+1}:\ldots:x_n]$, where $e_j\in\mathbb K^{n+1}$ the $j’$th unit vector is. Now I come to my question; How can I show that the images of $\tau_j$ overlay whole $\mathbb KP^n$ or mathematically spoken: $\mathbb KP^n=\bigcup_{j=0}^n \tau_j(\mathbb K^n)$ I think […]

projective hypersurface is a submanifold

I’m trying to prove that in $RP^2$, given a homogeneous polynomial $F$ of degree $k$, the hypersurface $Z(F)$ of the zeros of $F$ is smooth if $\frac{\partial F}{\partial x_0}$, $\frac{\partial F}{\partial x_1}$ , $\frac{\partial F}{\partial x_2}$ are not all simultaneously $0$ on $Z(F)$. What I tried to do is to show that $Z(F)$ is a […]

Lines in the projective plane

In my lecture notes we have the following: The set $$\mathbb{P}^2(K)=\{[x, y, z] | (x, y, z) \in (K^3)^{\star}\}$$ is called projective plane over $K$. There are the following cases: $z \neq 0$ $$\left [x, y, z\right ]=\left [\frac{x}{z}, \frac{y}{z}, 1\right ]$$ $$\mathbb{P}^2(K) \ni \left [x, y, z\right ] \to \left (\frac{x}{z}, \frac{y}{z}\right ) \in […]

Projection onto subspaces – point to line projection

In the following document about projection onto subspaces, the author is computing the transformation matrix to project a vector $b$ onto a line formed by vector $a$. Since the projected vector $p$ is on the $a$ line, therefore it can be expressed as $p = \bar{x}a$. The projection “line” which is the vector $b – […]

Projective plane division by non-concurrent lines

How many regions does n non-concurrent lines divide a Projective Plane into? (probably a standard problem, but I am having conflicting answers)

Cutting the $2$-dimensional real Projective Space

I have the following question. Let $M$ be a smooth manifold which is homeomorphic to $\mathbb{R}P^{2}$. If one cuts $M$ along a non-contractible path then $M$ should be homeomorphic to a closed disc, right? Why is this so? Ben

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups “arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$”. Is there some intuitive explanation for this connection? I understand that $SO(n)$ is the set of all rotations about the origin of $n$-dimensional Euclidean space $\mathbb{R}^n$ I understand that the […]

Ways of defining topology on $P^{n}(\mathbb{R})$

I was wondering if the following two ways of defining topology on $P^n(\mathbb{R})$ are the same and why? Since $P^n(\mathbb{R})$ is the quotient space of $\mathbb{R}^{n+1}$, define the topology on $P^n(\mathbb{R})$ to be the quotient topology ( i.e. the maximal topology that can make the quotient map $q: \mathbb{R}^{n+1} – \{0\} \to P^n(\mathbb{R})$ continuous, if […]

integral cohomology ring of real projective space

What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher’s book and Proving that the cohomology ring of $\mathbb{R}P^n$ is isomorphic to $\mathbb{Z}_{2}[x]/(x)^{n+1}$. For the graded module structure, it is obtained from the homology struture.

Type and number of points at infinity on Riemann surface

Consider a polynomial $P(z) = z^4 \in \mathbb{C}[z]$. Set-theoretically $P(z)$ has one root equal to zero. From algebraic point of view it has four roots: root zero has multiplicity four. Also we can’t draw a curve in $\mathbb{C}$ around one of such roots but not around the others. Now consider a Riemann surface $X$ given […]