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1answer
29 views

how to compute the distance between a matrix and its lower rank approximation?

I have a matrix $X$ and $Z$ a lower rank approximation of $X$ obtained using only few of the columns of $X$. I would like to have a measure of how distant are $X$ and $Z$. In particular I would like ...
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0answers
16 views

Normalize matrix so that its projection equals identity - which to use?

I wish to normalize a given matrix M, n by k, so that its projection matrix equals the identity. This is to speed up computations. The transformation/decomposition would look like this: Generate M by ...
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0answers
16 views

Saari's homographic conjecture and the actual definition of homography

By the wikipedia definitions found here and here, and especially by the definition implicit in this MSE post, it seems two images are homographic if they are renderings of the same set of points in ...
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0answers
16 views

Measures on $\mathbb{P}^1(\mathbb{C})$ and $\mathbb{P}^1(\mathbb{C}_p)$?

Is there a natural measure on the above spaces? Ideally, I would like a measure that is invariant under automorphisms of $\mathbb{P}^1$.
4
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1answer
130 views

Blow Up: Resolution of Singularity

For blow ups, I have worked only in $\mathbb{CP}^2$. Once I locate the base-point, say $[x,y,z]=[0,1,0]$, I go back to $\mathbb{C}^2$ by considering the chart $y=1$. I then proceed to blow up ...
1
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1answer
56 views

Some questions about complex curves in $\mathbb CP^2$

I would like to ask for some clarifications in the following questions about complex curves. My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a ...
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0answers
27 views

Morphisms of quasi-projective varieties

Let $Y\subseteq \mathbb{P}^n(k)$ be a quasi-projective variety. By Görtz, Wedhorn (page 32, Proposition 1.65) in order to show that $$h:Y\to \mathbb{P}^m(k), y\mapsto (f_0(y):\dots :f_m(y))$$ is a ...
2
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1answer
51 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
2
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2answers
38 views

Projective varieties and irreducibility

The "modern"(schematic) definition of a projective variety is the following: Let $k$ be an algebraically closed field. A projective variety over $k$ is a closed subscheme of $\mathbb ...
1
vote
1answer
34 views

What is the line going through points $(5, 5, 5), (2, 2, 2) \in \mathbb{R^3}$ when mapped it is mapped to a point in the real projective plane?

So the real projective plane is homeomorphic under a function $f$ to $\mathbb{R^3} - (0, 0 ,0)$. Hence lines in $\mathbb{R^3} - (0, 0 ,0)$ become points in the the real projective plane. So what is ...
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1answer
123 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
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1answer
14 views

Isomorphism between projective space and grassmanian $\mathrm{Grass}(1,n+1)$

I have to show that the projective space $\mathbb{P}^n$ is isomorphic to the grassmanian $\mathrm{Grass}(1,n+1)=\{V\subseteq\mathbb{R}^{n+1}:V\,\text{linear subspace,}\,\dim\,V\,=1\}$ as well as ...
2
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1answer
17 views

Vector space and its Projecctivized Space

Why is the co-dimension one subspaces are the points of $\mathbb P(V^{\vee})$. $V^{\vee}$ is the dual space of V and and $\mathbb P(V)$ is the projectivized space of V. $\mathbb P(V)= ...
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2answers
44 views

How to show that $f_* (\sigma)=\sigma$ where $f$ is mapping between projective spaces $\mathbb{R}\text{P}^3$

Suppose that $f:\mathbb{R}\text{P}^3 \to \mathbb{R}\text{P}^3$ is continuous mapping without fix points and let $\sigma$ be (some) generator of group $H_3(\mathbb{R}\text{P}^3)$. Prove that ...
0
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1answer
30 views

The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle

Let $z,z_1,z_2,z_3$ be four points on the extended plane. Their cross-ratio $(z,z_2,z_3,z_4)$ by definition is the image $Tz$ of $z$ under the Möbius transformation $T$ that sends $z_1,z_2,z_3$ to ...
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0answers
20 views

A little problem about $4$ points in $\mathbb P^1(\mathbb C)$

I have to solve the following problem: Consider a set $D\subseteq\mathbb P^1(\mathbb Q)\subset \mathbb P^1(\mathbb C)$ such that $|D|>3$. Then there exists a Moebius transformation $M$, and ...
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2answers
46 views

Enumerative projective geometry

I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, ...
3
votes
1answer
124 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
1
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1answer
17 views

The degree of a map between complex projective lines

Let $P$ and $Q$ be complex polynomials such that $\deg P=p$, $\deg Q=q$ and $\gcd(P,Q)=1$. How can I: show that $F(z)=\frac{P(z)}{Q(z)}$ defines a smooth map $\mathbb{C}P^1\to\mathbb{C}P^1$? ...
2
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1answer
186 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
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0answers
31 views

Dimensions of the cohomology groups of certain complicated space

Let contruct the space $X$. We take the complex projective space $\mathbb{C}P^2$, pick two points $p_1, p_2 \in \mathbb{C}P^2$ and remove two small, disjoint, open $4$-balls $B_j$ centered at $p_j$. ...
1
vote
1answer
23 views

How to define a “distance” from point to line in 3D projective space which is projectively invariant?

Since the concept of distance in Euclidean space is not invariant in projective space, that is , distance is invariant under Euclidean transformations but not under projective transformations, is it ...
0
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1answer
34 views

Matrix for orthogonal projection

Given was $v_1 = \begin{pmatrix} i\\0\\1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 0\\i\\1 \end{pmatrix}$. 1) I needed a orthonormalized basis $B$ of the sub-space built by $v_1, v_2$ and I got: $B ...
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0answers
14 views

Projective line bundles and blowing up

I'm trying to understand some facts about Chern classes. Looking up for some special examples I found that seems the total space of the projective line bundle $P(M\oplus M^{-1})\to \mathbb{CP}$ must ...
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1answer
42 views

Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$ where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
1
vote
1answer
21 views

Is this subset of $PSL(n,\mathbb{R})$ Zariski-closed?

For some non-identity element $[A]\in PSL(n,\mathbb{R})$ ($[A]$ being the class of $A\in SL(n,\mathbb{R})$) and linearly independent vectors $x,y\in\mathbb{R}^n$, let $[x],[y]$ denote the classes of ...
2
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1answer
89 views

What exactly does a Mobius Transformation do?

From what I understand, a Mobius transformation is of the form f(z) = $\frac{Az+D}{Cz+B}$ where A,B,C, and D may be real or complex What is f(z) doing to z exactly? And what are some of the ...
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1answer
29 views

How does a linear fractional function behave like a $2\times 2$ matrix?

So I did the math for this and got \begin{align*} A &= a_1a_2 + b_1c_2\\ B &= a_1b_2 + b_1d_2\\ C &= c_1a_2 + d_1c_2\\ D &= c_1b_2 + d_1d_2. \end{align*} My book does not talk ...
0
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1answer
93 views

What does it mean to call horizontal lines through O the “points at infinity” in real projective plane $RP^2$?

This is a picture from my book. I extended the line M to get a better idea of where $p_n$ is. It says the following: It is natural to call the horizontal lines through O the "points at infinity". ...
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0answers
30 views

Isometries and geodesics in projective plane using covering

We define a relation in the sphere by identifying the antipodal points, the quotient space obtained is the projective plane $\mathbb{P}^2$. Also, the quotient map ...
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0answers
30 views

A peculiar fact about 3-dimensional complex projective space

I'm working on a result for my master's thesis, that right now involves translating a proof I don't quite follow, to something that is a bit more in line with what I already know. We define ...
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0answers
42 views

Problem about $\mathbb{P}^3(K)$

Show that four skew lines in $\mathbb{P}^3$ have two transversals in common. I know that exist a quadric which contains three of the four lines....but i'm stuck EDIT: If the skew lines are ...
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1answer
26 views

Is there any definition for homogeneous rotations?

Most of the geometric transformations can only be represented into square matrices via homogeneous coordinates, e.g., translation and 3D rotations with axes not through coordindate system origin. ...
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0answers
28 views

A question about complex projective $n-$space

Let $\mathbb{P}^{n}(\mathbb{C})$ be the complex $n-$projective space and let $$ U_i=\{[x]=[x_0:\dots:x_n] \in \mathbb{P}^{n}(\mathbb{C}): x_i \ne 0\} $$ be a subset of $\mathbb{P}^{n}(\mathbb{C})$. I ...
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2answers
61 views

Finding singularities of a projective curve

For $w \in \mathbb{C}$ we define the projective curve $$p(x,y,z):= x^3+y^3+z^3+wxyz.$$ Now I have to find all $w \in \mathbb{C}$ for which the projective curve $p(x,y,z)$ is singular and show that ...
3
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0answers
25 views

Is there an embedding of projective varieties $\mathrm{Grass}(r,n)\hookrightarrow(\mathbb{P}^{n-1})^{\times r}$?

Let $k$ be an algebraically closed field, and let $r\le n$ be positive integers. Let $\mathrm{Grass}(r,n)$ be the projective variety of all $r$-dimensional planes in $k^n$. Notice that ...
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2answers
66 views

Elementary and purely Topological Proof of the non-triviality of Tautological Complex Line Bundle

I need some hints about the proof of the non triviality of the tautological complex line bundle, in a pure topological manner. Let $E$ be the t.c.l. bundle defined in this way $$ E= \{ (x,v) \in ...
0
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1answer
45 views

Is the projection map from direct product of projective $n$ space and projective $m$ space to projective $n$ space a closed map?

Is the projection map from direct product of projective $n$ space and projective $m$ space to projective $n$ space a closed map? I know that if $X$, and $Y$ are topological spaces with the product ...
2
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1answer
39 views

The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$

Let $X$ be irreducible algebraic set of projective n space. I am trying to show that: The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$, where $G(k,n)$ is the ...
0
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1answer
33 views

Projective Space

I have a question and I hope you can help me. I know that the complex projective space, for example ℂ ³ P, consists of the equivalence classes [z] = Z {α: α ∈ ℂ \ {0} and z ∈ ℂ ⁴}. I have a problem ...
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1answer
175 views

$\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$

I have to solve the following: Show that $\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$ for $n\geq 2$. I have done this with knowledge of homotopy-groups, by showing that ...
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0answers
31 views

Automorphisms of Complex Projective Space

Each automorphism $\mathbb{C}P^n\rightarrow \mathbb{C}P^n$ (where $\mathbb{C}P^n$ is regarded as a complex manifold) is induced by a linear map $\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$. I know ...
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0answers
24 views

Generalization of a projective plane?

In the area of finite geometry, a projective plane is an incidence structure of points and lines with the following properties: Every two points are incident with a unique line Every two lines are ...
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1answer
22 views

Why does it take 5 points to construct a projective frame in $\mathbb{R}^4$?

I am reading this answer http://math.stackexchange.com/a/186254/130408. The original question in that post is about deducing the projection matrix. But I have difficulties in understanding that ...
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0answers
41 views

An exercise in Gathmann's lecture notes about Projective spaces

Exercise 3.5.1 in Gathmann's lecture notes: Let $L_1$ and $L_2$ be two disjoint lines in $\mathbb{P}^3$, and let $P\in \mathbb{P}^3 \setminus (L_1 \cup L_2)$ be a point. Show that there is a ...
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0answers
44 views

How to find equations that define the image of an algebraic morphism?

Suppose we have a map $f:\mathbb{P}^n\rightarrow \mathbb{P}^m$ which is algebraic. What are the techniques to find the equations defining the image of f as a subvariety of $\mathbb{P}^m$? For example ...
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1answer
74 views

Computing the Todd class of projective space.

As an exercise I'm trying to verify that for $X=\Bbb{P}_k^n$, where $k$ is an algebraically closed field, we have $$\operatorname{td}(X)=\left(\frac{\epsilon}{1-e^{-\epsilon}}\right)^{n+1},$$ where ...
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1answer
35 views

How can I prove formally that the projective space is a Hausdorff space?

I want to prove the Hausdorff property of the projective space with this definition: Define $\mathbb{P}^n$, the real projective space of dimension n to be the set of 1-dimensional linear subspaces ...
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2answers
178 views

orthogonal projection and Cauchy Schwarz inequality

Show that if P is an orthogonal projection matrix, then $||Px||\le||x||$ for every x. Use this inequality to prove the Cauchy Schwarz inequality. I know that if P is an orthogonal projection matrix, ...
3
votes
1answer
77 views

Imagining the projective Space

I am trying to get used to work in the projective space. Therefore I wanted to know which tactics there are to imagine the projective space. $$\mathbb{P^n}(k):= (k^{n+1}\backslash \{0\})/k^{*}$$ I ...