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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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Fundamental group of the complement of $3$ disjoint hypersurfaces in $\mathbb{C}P^2$

Let $X$ be the union of $3$ hypersurfaces in $\mathbb{C}P^2$, then how to compute the $\pi_1(\mathbb{C}P^2\setminus X)$? What I know is the complement of a hypersurface in $\mathbb{C}P^2$ is $\mathbb{...
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smooth projective quadric $Q_{n,m}$ in $\mathbb{P}_n(\mathbb{R})$ with planarity $m$

How to show that smooth projective quadric $Q_{n,m}$ in $\mathbb{P}_n(\mathbb{R})$ with planarity $m$, dimension $n$ and $i=p-q$ (where $(p,q)$ is signature of the quadratic form) are related by $2m+i=...
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22 views

On the definition of degree of closed subschemes

$\underline {Background}$:We know that,for a projective variety $X \subset\mathbb{P}^{n}=(\mathbb{K}^{n+1}-{0})/\sim$ we define , degree($X$)=$(r!)$.(leading coefficient of the hilbert polynomial of ...
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1answer
21 views

A problem in multispace $\Bbb P^{n_1}\times \Bbb P^{n_2}\times …\times \Bbb P^{n_r}\times \Bbb A^m$

Consider the multispace $$M:=\Bbb P^{n_1}\times \Bbb P^{n_2}\times ...\times \Bbb P^{n_r}\times \Bbb A^m.$$Let $$f\in k[x_{11},...,x_{1n_1},x_{21},...,x_{2n_2},...,x_{r1},...,x_{rn_r},y_1,...,y_m]=k[...
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22 views

Projection map from projective space is not well defined?

Let $f: \mathbb{P^2} \to \mathbb{P^1}$ send $(x,y,z)$ to $(x,z)$. My book is telling me that this map is not well defined at $(0,0,1)$. How is this the case? Here $\mathbb{P}^1$ and $\mathbb{P}^2$ are ...
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1answer
58 views

Linear subspaces of projective space.

I'm following a basic course in Algebraic Geometry where the lectures are based on the first chapter of Algebraic Geometry by Robin Hartshorne. Our lecturer gave an additional advanced exercise after ...
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1answer
38 views

On a sheet of paper it is drawn a point $A$ and 2 lines $L_1$ and $L_2$ which intersect outside the sheet in a point B. [duplicate]

On a sheet of paper it is drawn a point $A$ and 2 lines $L_1$ and $L_2$ which intersect outside the sheet in a point B. How one can draw a line $AB$ using only a ruler?
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11 views

The Möbius Strip diffeomorphism

Are Möbius Strip and $\mathbb{R}P^2$ diffeomorphic? I really don't know how to approach this, I have tried to construct a diffeomorphism but couldn't get anywhere.
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35 views

On double points of a Projective space

$\underline {Background}$: Let, $ p\in Proj(K[x_0,....,x_n])$ . A double point at $ p\in Proj(K[x_0,....,x_n])$ is the scheme given by the square of the ideal (sheaf) of p. i.e we can consider it ...
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$W=\mathbb{P}(S)$ for some (k+1)-dimensional subspace $S$ of $\mathbb{k}^{n+1}$

Let $W$ be a subset of $\mathbb{P}_n(\mathbb{k})$. It is known that for every affine map $A$ (with which $W$ intersects) $A \cap W$ is a k-dimensional affine space. Is it true that $W=\mathbb{P}(S)$ ...
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1answer
70 views

Homology group on the complex projective space

I need to find the homology group of $$ X=l_1 \cup l_2 \cup ... \cup l_n \ \ \subset \mathbb{CP}^2 $$ where $ l_i $ are distinct projective lines. I think this is the direct sum of the homology ...
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37 views

What is the multidegree of a curve $C \subset \mathbb{P}^n \times \mathbb{P}^m$?

What is the multidegree of a curve $C \hookrightarrow \mathbb{P}^n \times \mathbb{P}^m$? I'm reading Notes on stable maps and quantum cohomology by W. Fulton and R. Pandharipande, and on page 14, ...
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60 views

Define the set $E(\mathbb{Q})$ of $\mathbb{Q}$-rational points on an elliptic curve

I'm a bit struggling with defining the set of $\mathbb{Q}$-rational points on an elliptic curve $E:\;y^2=x^3+ax^2+bx+c$ with $a,b,c\in\mathbb{Q}$. I'm actually guessing that If we let $K$ and $L$ be ...
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1answer
44 views

Projective Transformation for two distinct vertices and a linear objective function

I am trying to understand how to prove the following, which seems to be a quite useful insight in terms of linear optimization. Unfortunately, I have a hard time with projective geometry. I'd greatly ...
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57 views

The genus of a smooth algebraic curve in $(\mathbb{CP}^1)^2$

I came across this exercise in Riemann Surfaces by S. Donaldson: Let $Z$ be a smooth algebraic curve in $\mathbb{CP}^1\times\mathbb{CP}^1$. Let $d_1,d_2$ be the degrees of the projection maps ...
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1answer
28 views

Cross-ratio of points in the real projective plane

I would like to compute the cross-ratio of the points $A,B,C,D \in \mathbb{RP}^2$, in the projective plane, given by: $$ A=(0:1:2) \quad B=(1:2:3) \quad C=(2:3:4) \quad D=(3:4:5) $$ First I want to ...
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1answer
30 views

There cannot be a concept of parallelism in a homogeneous projective space?

Page 4 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: Affine geometry. We will take the point of view ...
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1answer
23 views

Vertical lines in the projective plane $P^2$

How do you get that two vertical lines in $P^2$ intersect at $(0 : 1 : 0)$ or how do you calculate it? If we look at two parallel lines, their point of intersection is at $(1 : s : 0)$ with s as the ...
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Tensor product of very ample line bundle with globally generated line bundle is very ample

I think I solved Exercise II 7.5 (d) from Hartshorne's Algebraic Geometry, but I don't know if I used the hypothesis. Those things alway leaves me doubtful if I made a mistake, so I would like to know ...
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23 views

Image of a line in $\mathbb{P}^2$ under Veronese embedding; a conicc [duplicate]

Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where $\phi(x_0,x_1,x_2)=(x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2,x_2^2)$. The problem asks to show ...
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Image of a line in $\mathbb{P}^2$ under Veronese embedding

Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where $\phi(x_0,x_1,x_2)=(x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2,x_2^2)$. The problem asks to show ...
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52 views

2 out of 3 property of fibrations?

Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a diagram in the category of topological spaces. If $g\circ f$ and one of $f,g$ are fibrations, can we conclude that so is the other? In this question it is ...
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1answer
68 views

The map of graded rings $k[w, x, y, z] \rightarrow k[s, t]$ induces a closed embedding $\mathbb{P}_k^1 \rightarrow \mathbb{P}_k^3$

Show that the map of graded rings $k[w, x, y, z] \rightarrow k[s, t]$ given by $(w, x, y, z) \mapsto (s^3, s^2t, st^2, t^3)$ induces a closed embedding $\mathbb{P}_k^1 \rightarrow \mathbb{P}_k^3$, ...
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24 views

Finding correct projective change of coordinates $M$.

Find a $3 \times 3$ matrix $M$ such that, under the change of varibles $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = M^{-1}\begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix},$...
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25 views

Property of Unirational variety

Let $X$ be an algebraic variety over field $k$ snd $n=\mathrm{dim}(X)$ . We assume $X$ is unirational. There exists $m \in \mathbb{N}$ and a dominant rational map $\phi : \mathbb{P}_k^m \...
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Can an algebraic variety be embedded in projective space ???

Let $X$ be an algebraic variety over field $k$ . $X$ can be embedded in a complete variety by Nagata's compactification theorem. Moreover, can we embed $X$ in a projective space $\mathbb{P}_k^n$ $?...
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1answer
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Kernel of ${\rm GL}(n,F)$ on ${\rm PG}(n-1,F)$ over a division ring $F$

I am reading Peter Cameron's note on Classical Groups and I got confused with Proposition 2.1 on page 14. I have no problem in proving that the elements in kernel are scalars. However, I don't ...
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1answer
70 views

Constant morphism between projective varieties

i'm currently working my way through Algebraic Geometry and stuck at Exercise 7.9 b) Let $f: \mathbb{P}^n \to \mathbb{P}^m$ be a morphism Prove: If $ n > m $ then $f$ must be constant I found ...
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Pencil and fundamental group

Let $f_i(x,y,z)$ be a homogeneous polynomial of degree $i$. Then for pencil $C_{a,b}=a(f_2)^3+b(f_3)^2$, we have a map $\phi:\mathbb{C}P^2\setminus B\to \mathbb{C}P^1$, where $B$ is the base locus of ...
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64 views

Independent projections

Suppose, we have a matrix $X\ (n\ \times\ n) $. There are r < n independent columns(therefore r is rank).And we project our vectors with $P\ (n\ \times\ n) $ operator = $I - E$ where E is the just ...
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1answer
30 views

Let $A$ be an operator, such that $A^{\dagger}A$ is a projector. Show that $AA^{\dagger}$ is also a projector

There is a hint to this problem which I don't know how to interpret. The hint is $$ \text{Hint: Show that} \quad A | \phi \rangle = 0 \leftrightarrow A^{\dagger}A=0. $$ Attempted solution(without ...
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1answer
56 views

Embed a weighted projective space into an unweighted projective space.

To show is the following. Let $X = P(a_0,\dotsc,a_n)$, $a_i \geq 1$ be a weighted projective space (that is $X = \operatorname{Proj} k[x_0,\dotsc,x_n]$, where $\operatorname{deg} x_i = a_i$). ...
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12 views

Triangle Inequality for Angles in Projective Space

I want to show that the angle between two lines through the origin in a (complex or real) inner product vector space $(V,\langle \cdot,\cdot\rangle)$ is a distance function which turns $\mathbb{P}V$, ...
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42 views

Disjoint Lines in Projective Space [duplicate]

I’m trying to prove the following result: Given $3$ pairwise disjoint lines $L_1$,$L_2$ and $L_3$ in $\mathbb{P}^3(k)$, we can find a change of coordinates such that $L_1=V(Z,T)$, $L_2=V(X,Y)$ and $...
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13 views

Transformation of sets defined by polynomial inequalities in a subspace to another subspace

Given a set defined by polynomial inequalites in one subspace of $\mathbb{R}^n$, is it possible to determine polynomial inequalites defining the set in another subspace of $\mathbb{R}^n$? The first ...
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1answer
29 views

Convergence of power matrix $ (I-\nu P)^m \to (I-P), \ for \ m \to \infty, $ with $0 < \nu < 1$

Suppose $P$ is a $n \times n$ projection matrix with two eigenvalues equal to one. Is it true that and how can I show that $$ (I-\nu P)^m \to (I-P), \quad for \ m \to \infty, $$ with $0 < \nu < ...
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0answers
21 views

Geodesics of the complex projective space

Is the complex projective space, a geodesic space? Is the complex projective space, a convex space? Let H be hyperplane of $\mathbb{C} P^n$, Is $\mathbb{C} P^n\setminus H,$ a bounded convex space?
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1answer
50 views

Fundamental group of $\mathbb{R}P^2$ in 2 models

I know that $\pi_1(\mathbb{R}P^2)\cong\mathbb{Z}_2$, but in the square model, I get that $\pi_1(\mathbb{R}P^2)=\langle a,b\colon abab\rangle$. These groups must be isomorphic, but I can't find the ...
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12 views

Probability before and after PCA projection

If there is a set of points generated from a multivariate normal distribution with mean and covariance matrix: mean=[1, 2]; covariance=[5, -2; -2, 3]; Data in ...
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13 views

Inclusion–exclusion principle in projective geometry

In the problems that I have to apply Grassmann in projective geometry, can I use the inclusion-exclusion principle? Consider the following problem: We consider three linear varieties of dimension ...
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1answer
95 views

Morphism to Projective Space $\mathbb{P}^m$

I'm going to prove following exercise from Bosch's "Algebraic Geometry and Commutative Algebra" (page 461): I'm struggle with proving that if $m < n$ then the morphism $\varphi: \mathbb{P}^n_R \to ...
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1answer
15 views

How locus of points of parallel lines in homogeneous coordinates, forms infinity?

In the diagram shown in link, what does writer mean when he says "locus of these points forms the line r". In the diagram "r" is curved, why is it called a line. I am facing difficulty in grabbing the ...
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27 views

Definition of an almost complex hyperplane in the projective space $\mathbb{C} P^n$.

Let J be an almost complex Structure in the projective space $\mathbb{C} P^2$. According to Duval a J-line in the almost complex projective space $\mathbb{C} P^2,$ is the almost complex analogue of a ...
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1answer
33 views

An injection yields an isomorphism in cohomology? $\Bbb RP^n \rightarrow \Bbb RP^{\infty}$.

In page 79 line 10 Hatcher made the claim that An embedding $\Bbb RP^{n-1} \rightarrow \Bbb RP^\infty$ induces an isomoprhism $$H^i(\Bbb RP^n, \Bbb Z_2) \leftarrow H^i(\Bbb RP^\infty, \Bbb Z_2)...
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2answers
95 views

How does the high school approach to classifying linear systems generalize to more variables, and why does it work?

If we have two linear equations in two variables, there may be $0$, $1$ or infinitely-many solutions. The standard approach to working out which case we're dealing with is probably to use Gaussian ...
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0answers
34 views

Spaces of matrices and projective spaces

Consider $\mathbb{C}^8$ i.e. 8 copies of the field of the complex numbers. I want to identify it with the set of matrices 2x4 $(z_{00},z_{01},z_{02},z_{03},z_{10},z_{11},z_{12},z_{13})$ now i have to ...
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1answer
23 views

Relationship between Affine definition of singular point and projective definition

Let $C : F(X,Y,Z) = 0$ be a projective curve given by a homogeneous polynomial $F \in \mathbb{C}[X,Y,Z]$, and let $P \in \mathbb{P}^2$ be a point. Prove that $P$ is a singular point of $C$ if ...
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1answer
38 views

A question about morphism of projective spaces

Consider the morphism: $$ f: (\mathbb{P}^2 -\{(0:0:1),(0:1:0) \} )\to \mathbb{P}^3 $$ Given by $f((x:y:z))=(x^2:xy:xz:yz)$, my problem is to find the closure of the image of $f$, my argument was: ...
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0answers
17 views

About the image of an affine variety into a projective space.

Assume to have a morphism $f: \mathbb{A}^n \to \mathbb{P}^m$, I want to compute the ideal of the the projective closure of $ f(\mathbb{A}^n)$. I think that in general the image of an affine variety ...
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0answers
19 views

What's the deal with projectively equivalent transformations?

Suppose that I have a homogeneous function from $\mathbb R^n$ to $\mathbb R$, where $ f(\alpha \mathbf{x}) = f(\mathbf{x}) $ for every scalar $\alpha$ and every vector $\mathbf x$ in $\mathbb R^n$. ...