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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Making measurements on a cuboid object from a single uncalibrated view

I have a cuboid box on which I have placed a reference object of known size. I am trying to find the dimensions of the box. I have already solved the problem with a calibrated view, but now I want to ...
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WLOG doubt: why can we assume that two disjoint linear subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$ are given by the following equations…

Let $H_1,H_2$ be two linear disjoint subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$. Let $(x_0:\cdots:x_n:y_0:\cdots:y_n)$ be homogeneous coordinates in $\mathbb{P}^{2n+1}$. My question is: ...
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Blowup of six points in $\mathbb{P}^2$

I have been looking at sequential blowups of six points $P_i$ in $\mathbb{P}^2$. In the general case, we can identify the blowup with a nonsingular cubic surface in $\mathbb{P}^3$ and under this ...
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28 views

Help required with question about closed unit ball in Hilberts space and proving the projection formula

I've this question that I intend to prove and any help will be appreciated
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40 views

Can $\mathbb{R}\mathbb{P}^2$ be embedded into an orientable 3-manifold?

We know that $\mathbb{R}\mathbb{P}^2$ cannot be embedded into $\mathbb{R}^3$, but is there an orientable 3-manifold where it is possible?
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Write $\mathbb{P}^3_{\mathbb{C}}$ as a union of disjoint lines

Is there a set $\Gamma=\{L \subseteq \mathbb{P}^3_{\mathbb{C}}: L \textrm{ is a projective line}\}$ such that every point $p \in \mathbb{P}^3_{\mathbb{C}}$ lies on exactly one line $L_p \in \Gamma$? ...
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76 views

Cohomology Group of $CP^2 \wedge CP^2$

Calculate the cohomology group of $CP^2 \wedge CP^2$ To do this, at first I am trying to calculate the homology group and then use Universal Coefficient Theorem. To do this, at first I have ...
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32 views

Convex functions up to reparametrization

I would like to know if there is a standard name for functions $f:[0,1]\to\mathbb R$ with the following convexity property: $$ \forall s<t<u\qquad f(t)\leq\max\{f(s),f(u)\}$$ (the fact that ...
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29 views

Divisors on Smooth Projective Curves

Hello fellow Mathematicians/Algebraic Geometer, very straight forward questions i) Explain concretely the DVRS $R$ with $k\subset R\subset k(t)$ where $k$ is an algebraically closed field ...
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42 views

Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
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Gluing together holomorphic functions on $\mathbb{P}^n$

The problem Let $U_j$ for $0\leq j\leq n$ denote the standard coordinate charts of the complex manifold $\mathbb{P}^n$. Fix $d\geq 1$ and assume we are given holomorphic functions $f_j:U_j\to ...
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Homemorphism from projective plane to S1 and Moebius strip

Let $h$ be a homemorphism from $S^{1}$ to the border of the Möbius strip $M$. Also, let $X$ be the quotient of the disjoint union of the closed unit disk $D^{2}$ and $M$ by the equivalence relation ...
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Gap in Hartshorne I can't fill

Page 142, Example 6.11.4. I've been trying to go through the details of the sentence The proof of (6.10) shows that if $f \in K$ is invertible at $Z$, then the principal divisor $(f)$ on $X - Z$ ...
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31 views

Point at infinity

My main question is can point at infinity be (0:0:0). Actually I know this question may be ridiculous, since (0:0:0) is not even in projective space. But when I finding the point at infinity of a ...
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61 views

Parallelizing lines

Let $n \geq 1$ be an integer, and $L_1,\ldots,L_n$ be $n$ lines in $\mathbb{R}^3$ which are pairwise disjoint. Is it possible to move all $n$ lines continuously so that they never cross, and so as to ...
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27 views

Calculating the dimension of a projective variety?

I am having difficulty determining the dimension of a projective variety in general. For example, I am confused about the dimension of the projective variety $X-Y=0$ in $\mathbb{P}^3$. I was ...
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Domain of rational map $\mathbb{P}^2 \to \mathbb{P}^2$

Let $\phi:(t_0:t_1:t_2) \mapsto (\frac{1}{t_0}:\frac{1}{t_1}:\frac{1}{t_2})$. I think that we cat extend $\phi$ to rational map $\hat{\phi}$ with domain:$\mathbb{P}^2-\{(1:0:0),(0:1:0),(0:0:1)\}$. How ...
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49 views

Universal Property of the Universal Line

In "An Invitation to Quantum Cohomology" by Kock and Vainsencher, they talk about "the universal line", which is defined as the variety $U=\{ (L,p)\in Gr(1,\mathbb{P}^r)\times \mathbb{P}^r | p\in L ...
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Connection between Chladni Plates and Projective Geometry?

Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
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42 views

Question about $\mathbb{P^1}$ - without loss of generality doubt

I want to show that: Given three distinct points $P_1,P_2,P_3 \in \mathbb{P}^1$ and three distinct points $Q_1,Q_2,Q_3 \in \mathbb{P}^1$, there is a unique isomorphism $f: \mathbb{P}^1 \rightarrow ...
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35 views

Triangle inequality on the projective space

Given a unit $n$-sphere $\mathbb{S}^n = \{x \in \mathbb{R}^{n+1} : \langle x,x \rangle = 1\}$, we define the set $\mathbb{P}^n = \{[x] : x \in \mathbb{S}^n\}$, where $[x] = \{-x, x\}$, and a function ...
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Gluing construction of the projective space scheme.

When constructing the projective space scheme $\mathbb{P}_R^n$ for a ring $R$, we may take the subrings $$ A_i = R\left[\tfrac{X_0}{X_i}, \ldots, \widehat{\tfrac{X_i}{X_i}}, \ldots, ...
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What this notation R^3 ∖ (0, 0, 0) means?

I was reading a "Projective Space" article on Wikipedia, when I came across this line "equivalent definition is the set of equivalence classes of $\mathbb R^3 \setminus (0, 0, 0),$ i.e. 3-space ...
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The 2 Charts of “Blowing up the Origin in $\mathbb{C}^2$ ”

Consider the algebraic curve $\mathcal{C}$ given by $f(x,y)=0$, where $(x,y)\in\mathbb{C}^2$. Suppose that the singular point of $f$ is $p=(x,y)=(0,0)$. The blow-up of $p$ is given by ...
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find $\overline{U}$ explicitly in terms of the polynomials defining $U$ (Over $\mathbb{P}^n$)

Let $U_0 \subset \mathbb{P}^n$ be the set defined by $U_0= \{(x_0:x_1:...:x_n)\in \mathbb{P}^n: x_0\ne 0 \}$ Consider the map $\phi: \mathbb{A}^n \to U_0$ given by: $$ ...
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Is there a projective metric on a projective space induced by a p-norm?

A projective metric on a projective space is a metric on the underlying set such that shortest path with respect to this metric are parts of entire projective straight lines. The 2-norm induces the ...
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Nonlinear Co-ordinate transformation in Complex Projective Space

If (x_0 ^2, x_1 ^2, x_2 ^2) denote co-ordinates of points in the Complex Projective Space P^2, what space does (x_0, x_1, x_2) span?
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Projecting point in 3d space onto a 2d view

If I have the following information: The coordinates in 3d space of a point(x, y, z) The dimensions of a 2d viewing window(width, height) The coordinates in 3d space of the center of that view(x, y, ...
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Projection onto subspaces - point to line projection

In the following document about projection onto subspaces, the author is computing the transformation matrix to project a vector $b$ onto a line formed by vector $a$. Since the projected vector ...
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Proving $F\#T\cong F\#K$

Let $K$ be Klein bottle, $T$ a torus with one hole and $F$ a surface which contains Möbius strip (as a subspace). Show that $F\#T\cong F\#K$ (and in fact if $P$ is the projective plane $P\#T\cong ...
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Is $K\mathbb P^n\cong K^n\mathbin{\dot\cup}K\mathbb P^{n-1}$ for all fields $K$?

I know that for $K=\mathbb R$, the statement $\mathbb R\mathbb P^n\cong\mathbb R^n\mathbin{\dot\cup}\mathbb R\mathbb P^{n-1}$ (where $\cong$ denotes set isomorphism) holds. Is the identity $$K\mathbb ...
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If I homogenized $f(z) = \frac{az+b}{cz+d}$, what would that look like and what would the process be to doing it?

I have a quick question. If I homogenized $f(z) = \displaystyle\frac{az+b}{cz+d}$, what would that look like and what would the process be to doing it? I am extremely new to homogenization and I and ...
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54 views

Möbius band inside projective plane

How can I see inside the projective plane the Möbius band? I need to know how the Möbius Band appears inside the projective plane. I know it is easy using identifications and algebraic topology. ...
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Determining if a set is projective or not

In $\mathbb{P}^3$ define the following sets: $$X=\{w_0w_1^2=w_2^2w_3-w_3^3\}\\Y_1=\{y_3=0\}\\Y_2=\{\sum_{i=0}^3 w_i=0\}\\Y_3=\{w_0+w_1+w_2+2w_3=0\}$$ Does the set $Z=X\cap Y_3\setminus((X\cap ...
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Help with Diagram of the Standard Lift of Projective Plane

I am posting here because I need help finding (or making) a visual aid for a presentation. I am giving a short presentation about Projective Geometry next week, and I am building a beamer for it. One ...
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integral cohomology ring of real projective space

What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher's book and Proving that the cohomology ring ...
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Mipoint between points in projective space

Is there a way to define the midpoint between points in projective space?
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Lines in the projective plane

In my lecture notes we have the following: The set $$\mathbb{P}^2(K)=\{[x, y, z] | (x, y, z) \in (K^3)^{\star}\}$$ is called projective plane over $K$. There are the following cases: $z \neq ...
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The classes are lines of $K^3$ that passes through $(0, 0, 0)$.

In my lecture notes we have the following: We consider $(K^3)^{\star}=K^3 \setminus \{(0, 0, 0)\}$ and we define the relation $$(a_1, b_1 , c_1) \sim (a_2, b_2, c_2) \Leftrightarrow (\exists ...
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The Cusp $w^2 + p(z,w)=0$ is desingularizable in the origin $O \in \mathbb{C}$

I have just studied a method in projective geometry over complex numbers on how to desingularize a curve in a point but i'm a little bit confused. I don't know the name of this classical method in ...
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Two discrete lines always intersect at a point

In my lecture notes we have the following: $K$ field Extension of the affine space. Relation between points and lines: Two discrete points define an unique line and two discrete lines always ...
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The quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open.

I'd like to show that the quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open, where I'm considering $\mathbb P^n$ as the quotient space of $\mathbb R^{n+1} \setminus \{0\}$ ...
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Is $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$?

Just curious, is it true that $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$? Here I'm writing $\mathbb{A}^1$ is affine space, and $\mathbb{P}^1$ projective space, both over an ...
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Comparing vector bundle degrees coming from different embeddings into projective space

This question is a follow-up to this recent question of mine: Comparing notions of degree of vector bundle In that question, the definition of the degree of a vector bundle is discussed — in ...
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40 views

Projective coordinates for point at infinity on elliptic curve

What is the unique characteristic of the projective coordinates of a point at infinity? I am specifically looking for a characteristic on (short) weierstrass curves. I know that the point at infinity ...
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Questions about the space of rays with initial point the origin endowed with the quotient topology.

I need to know if some properties about the topological space $\mathbb{R}^n/{\sim}$ are true, where $\sim$ is a equivalence relation defined by $a\sim b \iff \exists \lambda >0$ such that ...
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What is the domain of definition of $S_1/S_0$ on $\mathbb{P}^2$?

Consider the regular function given by $S_1/S_0$ on the projective sphere $\mathbb{P}^2$ over a field $k$ (We can assume algebraically closed, if it's needed for some reason). I'm just worried, is the ...
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65 views

Vector Field in a complex projective space

This question is motivated by this answer here. Let $\mathbb{C}P^{n}$ be a complex projective space. Let $X\in\Gamma(T\mathbb{C}P^{n})$, be a vector field. It seems, by the answer I got in the ...
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75 views

zeros of a certain vector field in $\mathbb{C}P^{n}$

Let's consider the complex projective space $\mathbb{C}P^{n}$ and let $X$ be a vector field with flow given by $X_{t}:\mathbb{C}P^{n}\rightarrow\mathbb{C}P^{n}$ such that ...
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Does there exist a regular map $\mathbb{A}^1\to\mathbb{P}^1$ which is surjective?

Suppose $\mathbb{A}^1$ and $\mathbb{P}^1$ are affine space and projective space. I'm not sure if it matters, but I don't mind if we assume that we're working over algebraically closed fields. I'm ...