# Tagged Questions

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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### Punctured complex projective space

Let $\mathcal{P}\mathbb{C}^{n}$ be the complex projective space of $\mathbb{C}^{n+1}$, and let $B=\{\mathbf{e}_{1},\cdots,\mathbf{e}_{n+1}\}$ be a basis in $\mathbb{C}^{n+1}$. I would like to ...
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### Which line bundle has transition function $\psi_{12}([z_1,z_2])=\frac{z_1/z_2}{|z_1/z_2|}$?

We know that the complex line bundles over $\Bbb{CP}^1$ are classified by the integers. Each is isomorphic to one of the bundles $\mathcal{O}(n)$ for $n\in\Bbb Z$, where $\mathcal{O}(-1)$ is the ...
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### Effect on eigenvalues of a projection matrix when removing its main diagonal?

Do you know what happens to a projection matrix when you remove the main diagonal? Consider $P=Z(Z'Z)^{-1}Z'$ and form P-diag(P-{i,i}). Now, when you eigendecompose $P$ we get zeroes and ones as ...
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### Dimension of the fiber of a projective variety

When looking at this I found the theorem Given a surjective map of quasi-projective varieties $\phi:X\rightarrow Y$ and $y\in Y$, we have that dim$(\phi^{-1}(y))=$dim$(X)$-dim$(Y)$. First I was ...
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### Projective space, immersion, embeding

How can I check if the map $F:\Bbb{P}^2\to \Bbb{P}^5$, given as follows $F([x,y,z])=([x^2,y^2,z^2,xy,yz,zx])$, is smooth, an immersion or an embedding?
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### Extension of projection from a point to a Blow Up

I feel like there's something obvious I'm missing here, and I'm not looking for a whole answer, but rather just a pointer in the right direction. Suppose you have the projection from a point ...
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### Equivalence of a vector bundle being trivial on $\mathbb{P}^1$

I am looking for various statements about a vector bundle $E$ of arbitrary rank being trivial on the complex projective line, $\mathbb{P}^1$. In particular, some arguments about cohomology would be ...
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### Projective Linear Group and Projective Space

my aim : Why $GL(V)/Z(GL(V))$ is termed as projective general linear group? The reason seen in books says This group acts on the projective set/space faithfully. But, there can be many groups acting ...
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### Are $\mathbb{A}^n (k ) = k^n$ and $\mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles?

Are $\mathbb{A} (k ) = k^n$ and $\mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when $k$ is a domain ? Thanks in advance for your help.
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### Obtaining the Fundamental Polygon of $\mathbb{R}P^2$

On this page, Wikipedia shows, under the "Examples" heading, the fundamental polygons of the Sphere and the Real Projective Plane. Can we obtain the latter diagram from the former? I thought that this ...
If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...