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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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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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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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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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, ...
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 \$(...