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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under change of coordinates the variety $Z(H_1,..,H_r)$ becomes $Z(x_1,…,x_r)\subset \mathbb{P}^n$.

A set $V\subset \mathbb{P}^n$ is called a linear subvariety of $\mathbb{P}^n$ if it's the zero locus of $r$ homogeneous and linear, i.e $V=Z(H_1,...,H_r )$ where each $H_i$ is a form of degree 1. I ...
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Weighted projective space

Here is one example from Cox's lecture notes: I really don't know how to use $M$ to define an automorphism. Intuitively, I can rescale the first coordinate to $1$ and I think the isomorphism is ...
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Why is $\mathbb{P}(Sym_2(\mathbb{C}^2))$ isomorphic to $\mathbb{P}^2(\mathbb{C})$?

Let $Sym_2(\mathbb{C}^2)$ denote the space of symmetric 2-tensors on $\mathbb{C}^2.$ I want to understand why is $\mathbb{P}(Sym_2(\mathbb{C}^2)) \cong \mathbb{P}^2(\mathbb{C})$. Any help please?
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Divisors corresponding to hypersurfaces in Projective space

I'm looking at hypersurfaces on $\mathbb{C}\mathbb{P}^2$. That is, the zero set of an irreducible homogeneous polynomial $f(x_0, x_1, x_2) = 0$. This corresponds to a divisor, $D_f$, let's say which ...
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How to prove: for every two complementary subspaces there exists a projector

In Trefethen and Bau's book, Computational Linear Algebra, in the Projections chapter I've come across the following statement: Let $S_1$ and $S_2$be two subspaces of $\mathbb{C}^m$ such that $S_1 ...
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$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the ...
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2 dimensional representations of v dimensional points based on distance

I have a set of points $x_1, \dots, x_n \in \mathbb R^v$ I have a measure of the distance between each one of these points $D \in \mathbb R^{n\times n}$ where $D_{i,j}= distance(x_i, x_j)$ I would ...
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cohomology ring of a subspace of real projective spaces

I learned $H^*(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[a]/(a^{n+1})$, $|a|=1$, in topology class, when studying cell complex and cohomology. Now I want to find the cohomology ring ...
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Curvature for tautological bundle of projectivation

I'm trying to compute locally the curvature of $\mathscr{O}_{\mathbb{P}(E)}(-1)$, where $E\to X$ is a holomorphic bundle. The covering is given by $U_i\times \mathbb{P}(E)_j$, where $\{U_i\}$ is a ...
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Is the action of the matrix UU^(t) always a projection? What can I say about I - UU^t?

The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal ...
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Uniqueness of a projective transformation

Just as there exists a unique projective transformation that takes three points in $\mathbb{CP}^1$ to three other points in $\mathbb{CP}^1$, how many points do I need for the corresponding question in ...
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Model for a Smooth Curve

If $K$ is a finite Galois extension such that $\mathbb{F}_q(x)$ such that $K\cap\bar{\mathbb{F}}_q = \mathbb{F}_q$ then there exists a smooth projective curve $C$ such that $\mathbb{F}_q(C) = K$. My ...
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Orthogonal projection matrix P onto the range of a 3x2 matrix

I have a 3x2 matrix A = {{1,-1},{2,-1},{3,1}}. I need to find the orthogonal projection matrix P onto the range of A. I know that the orthogonal projection is the outer-product / inner-product, that ...
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Prove $\Bbb C^2 \setminus \{0\}/\Bbb C^*$ is homeomorphic to $(\Bbb C^2 \setminus \{0\})/_{\sim f}$, where $f=\frac{qi\bar{q}}{|q|^2}$, $q$ quaternion

The equivalence relation $\sim f$ is defined s.t. $q_1 \sim q_2$ iff $f(q_1) = f(q_2)$. I am having problems to start. First I have problem on understanding $\Bbb C^2 \setminus \{0\}/\Bbb C^*$. I ...
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Find a projective variety $Z$ and closed subsets $X,Y \subseteq Z$ with $\dim(X)+\dim(Y) \geq \dim(Z)$ and $X \cap Y = \varnothing$

I am trying to find a projective variety $Z$ and closed subsets $X,Y \subseteq Z$ with $\dim(X)+\dim(Y) \geq \dim(Z)$ and $X \cap Y = \varnothing$. However, all my attempts failed. In fact, we ...
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Point at Infinity of E.C. in Jacobian Coordinates

I am reading some notes about elliptic curves right now and the author mentions the alternative Jacobian projective coordinates, where one establishes the equivalence $(x,y,z)\sim (\lambda^2 x, ...
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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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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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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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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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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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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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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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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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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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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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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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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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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 ...