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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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Algebraic Sets in Projective Space

I have two homogeneous polynomials $f$ and $g$, and I want to decompose the vanishing set $\mathbb{V}(f,g) \subset \mathbb{P}^2(k)$ into irreducible algebraic sets. To start, I have factored $f$ and $...
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Find a projective change of coordinates that maps the (projective) line L to the (projective) line G

Let $L$, $G \subset \mathbb{P}^2$ be lines. Show that there exists a projective change of coordinates $T$, such that $T(L)=G$. This is how we defined a projective change of coordinates in $\mathbb{P}...
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Computing Fubini-Study metric from the formal definition

Definition: the Fubini-Study metric $g_{FB}$ on $\mathbb{CP}^n$ is the only metric which makes the projection $\pi:(\mathbb{S}^{2n+1},g)\to(\mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is ...
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1answer
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Submanifold of real projective space

Would you like to tell me how to prove that $\{[x_{0}:x_{1}:x_{2}]: x_{0}x_{1} + x_{2}^{2}\} \subset \mathbb{R}\mathbb{P}^{2}$ is a submanifold (of $\mathbb{R}\mathbb{P}^{2}$)?
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Example for intersection, multiplicity and tangents for projective plane curves

We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, ...
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Proposition 1.1 of A Royal Road to Algebraic Geometry (Holme)

I've been reading Holme's A Royal Road to Algebraic Geomtery, but I can't seem to prove the first proposition. It is stated (on page 8) as: Proposition 1.1 Given $n+2$ points $P_1,P_2,\ldots,P_{n+2}\...
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Non split real form of projective space

On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this ...
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1answer
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“line at infinity” in projective plane

("Algebraic Geometry: A Problem Solving Approach" by Thomas Garrity) I am struggling with Exercise 1.4.12.1 in the above, which I quote with some context: Here is my intuitive thinking: (a) lines ...
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1answer
31 views

Cohomology of Projective Space $\mathbb{PR}^n$ with Coefficients in $\mathbb{Z}/2$

We know the cohomology ring with coefficients in $\mathbb{Z}/2$ of projective real space $\mathbb{PR}^n$ is $$H^*(\mathbb{PR}^n, \mathbb{Z}/2) = \mathbb{Z}/2[X]/(X^{n+1})$$ with graduated generator ...
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Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies ...
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Proving that a point is the result of only two lines intersecting and a line is the result of only two points being aligned

Let $S_0$ be a set of four points in the real projective plane such that any three points of $S_0$ are not aligned. Let $L_0 := \emptyset$. For every integer $n \ge 1$, we define the following: ...
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Projective subspace of same dimension

I'm looking at families of curves in $\mathbb{P}^2$ (over $\mathbb{C}$), specifically the set $\mathcal{L}_d$ of projective curves defined by a homogeneous polynomial $P \in \mathbb{C}[x_0,x_1,x_2]$ ...
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1answer
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Projective/ Finite Geometric Basics!

I'm taking intro to coding theory and am having some trouble understanding the basics of Projective Geometry, since our text does not give it much discussion. Namely, if PG(r-1,q) is the set of all ...
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How to check for a projective space?

When I have a division ring commutative its pretty straight forward! But when its not commutative then I'm stuck, it's possible that in this case its not a projective plane? Can someone give a ...
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1answer
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Proving Hopf's Fibration $\pi: \mathbb{S}^{2n+1}\to\mathbb{CP}^n$ is a submersion

Prove that the following map is a smooth, surjective submersion: \begin{align*} \pi:\mathbb{S}^{2n+1}&\to\mathbb{CP}^n\\ (x_0,y_0,...,x_n,y_n)&\mapsto [x_0+iy_0:...:x_n+iy_n] \end{align*} ...
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If All the Points Lie On a Plane, Then Why Does the Linear Mapping Reduce to …?

I previously asked a question with regards to what the matrix $\mathrm{H}_{3 \times 3}$ is/represents in the following textbook excerpt: In applying projective geometry to the imaging process, it ...
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1answer
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Projective Transformations: “If all the points lie on a plane, then the linear mapping reduces to …”

Page 7 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following: In applying projective geometry to the imaging process, it is customary to model the world as a ...
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Textbook Error? “, … and singling out the line at infinity in *the image* or the plane at infinity in *space* when that becomes necessary.”

Page 3 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following: In computer vision problems, projective space is used as a convenient way of representing the ...
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1answer
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Is there a holomorphic diffeomorphism of $\mathbb{C}P^{2n+1}$ without fixed point?

Is there a holomorphic diffeomorphism $f:\mathbb{C}P^{2n+1}\to \mathbb{C}P^{2n+1}$ without fixed point?
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1answer
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Cohomology of tensor product of pullback in $\mathbb P^1\times\mathbb P^1$.

Let $X=\mathbb P^1\times\mathbb P^1$, and let $\pi_1$ and $\pi_2$ be the projection maps. For each $a,b\in\mathbb Z$, we have a sheaf of $\mathcal O_X$-modules $\mathscr F_{a,b} = \pi_1^*\mathcal O(a)\...
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The 2-Dimensional Projective Space $\mathbb{P}^2$ Has the Basis with *Three* (?) Vectors $\mathbf{e}_i$, $i = 1, \dots, 3$?

My textbook says the following: Consider a set of basis vector $\mathbf{e}_i$, $i = 1, \dots, 3$ for a 2-dimensional projective space $\mathbb{P}^2$. For reasons to become clear, we will write the ...
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De Rham cohomology of $\mathbb{RP^n}$

I have to calculate the De Rham cohomology of $\mathbb{RP^n}$ using the Mayer-Vietoris sequence. I first started by considering $\mathbb{RP^n}=S^n/\sim $ where $\sim$ is the antipodal identification. ...
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All the possible configurations in $\Bbb P^4$ of two planes whose sum is $\Bbb P^4$ and a line

Consider the real projective space $\Bbb P^4$ and let $\alpha,\beta$ be planes such that $\alpha+\beta=\Bbb P^4$. What are the possible configurations of $\alpha, \beta$ and a generic line $\ell$? ...
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1answer
31 views

Choice free method to define immersions into Projective Space

Let $X$ be a variety and $\mathcal{L}$ be a very ample line bundle on $X$. Suppose $H^0(X,\mathcal{L}) = \langle s_0,...,s_n \rangle$ then there is an immersion into projective space: $$ X \...
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1answer
26 views

$\mathbb C P^1\cong (D^2\times\{1\}+ D^2\times \{-1\})\big /_\sim $

I need to proove a bigger result that $\mathbb C P^1$ is homeomorphic to $S^2$. For that I have already showed $$S^2\cong (D^2\times\{1\}+ D^2\times \{-1\})\big /_\sim$$ where $(z,1)\sim (z,-1) \iff z ...
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How to show that $SO(3)$ and $\mathbb{R}P^3$ are dffeomorphic?

Pardon me for repeating a question, but I am not able to justify one aspect of this proof that $SO(3)$ and $\mathbb{R}P^3$ are diffeomorphic, and I was hoping that someone could clarify it for me. ...
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Show the one-point set $\{a\}$ is a projective variety, and compute explicit generators for the ideal $I_p(a)⊴K[x_0,…,x_n]$.

I think I have it, but I'm also new to this so I'm looking for verification. To be a projective variety, $\{a\}$ needs to be the zero locus of a set $S$, meaning $\{a\}=V(S)=\{x∈P:f(x)=0 \text{ for ...
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Morphism from $\mathbb{C}P^1$ to affine space

I am interesting the morphisms from $\mathbb{C}P^1$ to $\mathbb{C}^n$. I intuitively think that there should be only one nontrivial map, which is just embedding when $n>1$, and there is no ...
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Understanding projective spaces as generalized flag varieties

Briefly, is there an intuitive way to understand the relation $\mathbb{P}^n = \frac{U(n+1)}{U(1)\times U(n)}$ ? For instance, can one relate it to the usual definition of complex projective spaces as ...
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Prove check total space canonical bundle (vector bundle of projective space)

I have written a proof for the following question, but I'm not sure if I missed some subtilities or that I made some mistakes in my notation. Prove that $$K_\mathbb{R} = \left\{ \left( (y_0,\ldots, ...
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Critical Points of Polynomial Map $f: \mathbb{PC}^1 \to \mathbb{PC}^1$

Let $P(x):= a_nz^n +a_{n-1} z^{n-1} + ... + a_0$ be a polynomial with degree $deg(P) \ge 1$. One know that the map $P: \mathbb{C} \to \mathbb{C}, x \mapsto P(x)$ induced by this polynomial can be ...
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The symmetry group / isometry group of the complex projective space

question: For the complex projective space of $n$-complex dimensions, $$\mathbb{P}^n,$$ what is the symmetry group / isometry group of this complex projective space $\mathbb{P}^n$? Attempt: ...
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1answer
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Compute the degree of a particular morphism

Let $\mathbb P^n=\mathbb{CP^n}$ be complex projective space. Let $H^0(\mathcal O_{\mathbb P^n}(d))$ be the group of homogeneous polynomials of degree $d$, and denote its dimension by $N(d)$. Consider ...
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3answers
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Definition of sphere without using a metric

In differential geometry the $n$-sphere $S^n$ seems to be always defined as the set of points in $\mathbb{R}^{n+1}$ with distance $1$ from the origin. I am interested in a more topological definition, ...
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2answers
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Polynomial Forms on Dual Projective Spaces

Let $E$ be a finite dimensional complex vector space. Let $\mathbb{P}(E)$ be the projective space of lines through the origin of $E$. Fulton, in his book "Young Tableaux", then defines the $\textit{...
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From root and weight lattices of SU(N) to $\theta$-functions as sections of a line bundle and $CP$-space

I have troubles to digest the following messages/discussions in the following work in p.10-12; Which construct a map from the moduli space of flat connections $M_{\rm flat}=\mathbb{E} / {\mathfrak S}...
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2answers
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How to introduce a CW structure on RP^n?

My first course in topology is going extremely fast, and does not seem like rigorous mathematics. Last lecture, we were given the definition of CW-structures, but did not do any examples. Yet we were ...
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1answer
63 views

Nonsingular projective curve model corresponding to $y^2 = x^4+1$

Consider the affine curve $C_1 = V(y^2 - (x^4+1)) \subset \Bbb A^2_k$. In the answers to this question, they claim that there is a (unique?) nonsingular projective curve $C_2$ corresponding to $C_1$ (...
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1answer
51 views

A vector field on the projective plane with one critical point, a sink

I'm studying (for my own, this is not a homework question) vector fields on the real 2d projective plane. Is there a continuous vector field on $\mathbb R P^2$ (the 2d-sphere in $\mathbb R^3$ with ...
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$\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$ [closed]

I was stuck by reading this figure: It looks that $\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$ are somehow related. Are there some easier explanations from math ...
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1answer
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Closed oriented even dimensional manifold with only three non-zero Betti numbers.

The complex and quaternionic projective planes are the examples of a closed oriented even dimensional manifold with exactly three non-zero Betti numbers. For more example see the paper ''Rational ...
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Is the dehomogenisation of smooth homogeneous polynomial smooth?

If $F\in\mathbb{C}[x_{1},\ldots,x_{n+1}]$ is a smooth homogeneous polynomial, is it true that the dehomogenisation $F(x_{1},\ldots ,x_{n},1)\in\mathbb{C}[x_{1},\ldots,x_{n}]$ is also smooth? Smooth ...
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1answer
38 views

Projective classification of quadric from affine classification

Currently I'm taking a first course on Projective Geometry and I'm working on the following problem : Given a homogeneous polynomial $F$ of degree $2$ and $n+1$ variables we consider the quadric ...
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2answers
62 views

Induced homomorphism between fundamental groups

Let us consider the closed disk $\overline{\mathbb{B}(0,1)} \subsetneqq \mathbb{R}^2$. Let moreover $\mathbb{RP}^2:=\overline{\mathbb{B}(0,1)}/\sim$, where the equivalence relation $\sim$ identifies ...
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How do we know that the only polynomials $f$ satisfying $f(\lambda x_0,\ldots,\lambda x_n) = f( x_0,\ldots,x_n) $ are the constant polynomials?

In algebraic geometry one proves that the affine coordinate ring of $\mathbb{P}^n$ is trivial by using that the only polynomials $f$ satisfying $f(\lambda x_0,\ldots,\lambda x_n) = f(x_0,\ldots,x_n) $ ...
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0answers
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The Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $

I have a small question about the Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $ appearing in page : $ 119 $ of Daniel Huybrechts's book intiteled : Complex ...
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2answers
122 views

Calculating $\mathbb{R}P^1$ fundamental group.

Well, I am trying to use the fact that $S^1$'s fundamental groups is free and generated by on element ($\mathbb{Z}$), denoting $\pi_1(S^1) = \langle [\gamma] \rangle$. When $\gamma$ is a loop starts ...
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2answers
63 views

Homotopy type of a disk and homotopy type of a sphere.

According to the following link : Calculation of de Rham complex for real projective space , $ \mathbb{P}^d $ is devided in two open sets : $ U = \{ \ [ x^0 : \dots : x^d ] \ | \ x^d \neq 0 \ \} $ and ...
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0answers
63 views

How to compute $ H^{\bullet} (\mathbb{P}^n (\mathbb{R}), \mathbb{Q}) $, using Mayer Vietoris sequence?

How do we compute the Cohomology algebra of the real projective space: $ H^{\bullet} (\mathbb{P}^n (\mathbb{R}), \mathbb{Q}) $, by induction, and by using Mayer Vietoris sequence ? Thanks in advance ...
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1answer
52 views

Proof that every codimension 1 subvariety of $\mathbb{P}^n$ is $V((f))$

I've been told that Theorem. Every codimension 1 subvariety of $\mathbb{P}^n$ is $V((f))$, where $f$ is some prime homogeneous polynomial. I'm under the impression that this is true over any ...