# 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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### Uniqueness of the decomposition of rank 2 holomorphic line bundle over $\mathbb{CP}^1$

Let $\mathcal{O}(1)$ be the hyperplane bundle over $\mathbb{CP}^1$ and $\mathcal{O}(n)=\mathcal{O}(1)^{\otimes n}$. I knew that any rank 2 holomorphic line bundle over $\mathbb{CP}^1$ is isomorphic to ...
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### Čech cohomolog of a good cover of the real projective plane

I was reading the Bott/Tu book and trying compute the Čech cohomology of a good cover of the real projective plane (exercise 9.10). The nerve of the cover is dipicted as below: we glue the opposite ...
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### Kahler-Einstein metric on complex projective space

I think this question may be well-known to the experts; or someone may have already asked the following question in this website. Since I couldn't figure it out myself and I couldn't find a related ...
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### Topology of the projective space

I am studying the fundamental group and I am facing a problem with the projective space: we know that, by definition, the real projective space $P^m(\mathbb{R})$ is the quotient of $R^{m+1}-\{0\}$ ...
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### Quadric equation in Projective Geometry

It is well known that 5 points in $\mathbb{R}^2$ define a conic. While studying the book Geometry II by Marcel Berger, I came across a theorem stating that 9 points in $P\mathbb{R}^3$ define a quadric....
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### What's the point of the local zeta function?

I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see ...
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### Finite Order (in Homology Group) Implies Non-Orientability? (Intuition)

The motivation for this question comes from this post ("Where does the term "torsion" in algebra come from?"). To keep it brief: Given an element of finite order $\in H_n(X)$ (...
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### Which representation of $\mathrm{SL} (2; \mathbb{C})$ does the tautological bundle on $\mathbb{CP}^1$ correspond to?

If I understand correctly, by Borel-Weil-Bott, for each weight of $\mathrm{SL} (2; \mathbb{C})$ there is a corresponding (holomorphic) line bundle / invertible sheaf on $\mathbb{CP}^1$. Denoting the ...
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### Intersection multiplicity of two curves embedded in the projective plane

I encountered the following problem and hope to receive guidance from everyone. Let $\phi$ maping from $C[x,y]$ into $C[x,y,z]$ turning a polynomial $f(x,y)$ into $F(x,y,z)= z^{deg f}f(x/z,y/z)$. ...
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### Difference between the sphere and RP^2 in the generators and relations of $\pi_1$

Here are two drawings we had that I did not quite well understand why they are correct: Here are my questions: 1-I think the drawing of $RP^2$ is meant to be a circle not a disk, am I correct? 2- ...
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### Set is equal to its closure in projective space [duplicate]

I'm trying to determine some criteria for when the closure of a closed irreducible set $X$ of $\mathbb A^n$ is equal to the set itself. That is if $\overline{X}$ is the closure of $X$ in $\mathbb P^n$ ...
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### Irreducible components of an affine cone are affine cones [duplicate]

We have shown that for an affine cone of a nonempty subset $X$ of $\mathbb P^n$, i.e $C(X)$, that the irreducible decomposition of $C(X)$ given by $Y_1 \cup \dots \cup Y_m$ has that $Y_i = C(X_i)$ for ...
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### How to think about projective geometry problems

This semester I am taking an introductory course on projective geometry. Even though I believe I understand the different theoretical concepts correctly (supplementary varieties, Grassmann Formula, ...
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### If $\phi:V_1\rightarrow V_2$ is a morphism of varieties then $V_1\cong \phi(V_1)$

I am reading Silverman's The Arithmetic of Elliptic Curves. I am wondering if with the definition of morphism he gives, we can conclude that if $\phi:V_1\rightarrow V_2$ is a morphism of projective ...
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### Why $\mathbb{CP}^1 \cong S^2$? [closed]

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 $\mathbb{CP}^1 \cong S^2$ ...
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### Parallel lines diverge behind the observer in projective geometry

Of course parallel lines converge at a point at infinity in projective geometry, but visually, they appear to diverge as one gets closer and closer to the start of one's vision, i.e. they diverge ...
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### Tangent space of $\mathbb{P}(V)$
I encountered this problem studying the local period map and I'm wondering how to solve it. I would like to prove that, given $V$ a complex vector space and $W \subseteq V$ a one-dimensional subspace, ...
$\mathbb{CP}^1$ can be formed from $\mathbb{C}^\times = \text{GL}_1$ by gluing $\mathbb{C}$ by itself along $\mathbb{C}^\times$, a pushout of $1/z,z : \mathbb{C}^\times \rightarrow \mathbb{C}$. I am ...