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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How would one define polynomials over the projective line $P_K^1$

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form ...
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K-theory of projective space

Is there any way to prove that the twisting sheaves $\mathcal{O}(K)$ generate the algebraic K-theory of projective space without actually using any K-theory machinery (e.g. Bott periodicity)? Like for ...
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Intuitive argument for the transitivity of $PGL(n+1)$ acting on $\mathbb{P}^n$

In spherical geometry we can consider the action of $\operatorname{O}(n+1)$ on the unit sphere $\mathbb{S}^n$. It's easy to see that this action is transitive, because for any two points $x,y \in ...
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Affine varieties over finite fields

I read in this paper (http://www.math.iitb.ac.in/~srg/preprints/Chandigarh.pdf) that the following set is an affine variety: $V_f=\{(t_0,...,t_N)\in \mathbb{F}_p^{N+1} : f(t_0,...,t_N)=0 \}$ where ...
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Vanishing set of a homogeneous polynomial containing projective lines

I have a few questions about the following exercise. Let $K$ be an algebraically closed field. Let $F \in K[X_0,...,X_N]$ be a homogeneous polynomial of degree $ m > 0 $ and let $Z = V(F)$. Let ...
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2answers
73 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
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Proof of the Inscribed Angle Theorem

I want to give a proof of the Inscribed Angle Theorem by using the Laguerre formula. Let $C$ denote the circle. Take three different points $A,B$ and $P$ on $C$. Write $a := \overline{AP}$ and $b:= ...
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1answer
38 views

Homotopy type of intersection of complement of hyperplanes in projective space.

Let $U_i = \{x=(x_0 :… :x_n) \in \mathbb{P}^n(\mathbb{C}); x_i \neq 0 \}$ be the usual trivialization of the complex projective space. I have been trying to compute the homotopy type of all the ...
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Complex Projective Space as a Quotient of a Disc

I am reading Hatcher's book and I have a problem understading how the complex projective space $\mathbb CP^n$ can be realised as a quotient of $D^{2n}$ (page 7) Let me briefly outline his arguments ...
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1answer
27 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
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1answer
33 views

Real Projective Space Homeomorphism to Quotient of Sphere (Proof)

I need to construct a function $f : (\mathbb{R}^{n+1}-\{0\})/{\sim} \to S^n/{\sim}$, by $$f ([x]_{\mathbb{RP}^n}) = \left[\frac{x}{\|x\|}\right]_{S^n/{\sim}},$$ where $S^n = \{ x \in ...
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1answer
33 views

Given a point $P$ and a hyperplane $H$ in $\mathbb{P}^n$ such that $P \in H$, there is $T$ linear such that $T(P)=(0:\cdots:0:1)$ and $H:X_0=0$

Show that given a point $P$ and a hyperplane $H \subseteq \mathbb{P}^n$ such that $P \in H$, there is a linear transformation $T$ such that $T(P)=(0:\cdots:0:1)$ and $H$ is given by the equation ...
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51 views

Show that any quadric in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$

Show that any non-singular irreducible quadric in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ I know that every non-singular and irreducible quadric in $\mathbb{P}^3$ can ...
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Show that there exists a coordinate system in $\mathbb{P}^n$ such that $P_0=(1:0:\cdots:0),\ldots,P_n=(0:0:\cdots:1),P_{n+1}=(1:\cdots:1)$

Proposition: Let $P_0,\ldots,P_{n+1}$ be $n+2$ points in $\mathbb{P}^n$ such that every $n+1$ are in general position. There exists a coordinate system in $\mathbb{P}^n$ such that ...
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Visualisation of $\mathbb{P}^1(\mathbb{Z}/6\mathbb{Z})$

For some exercise I required the fact that there are $p + 1$ lines in $\mathbb{F}_p \times \mathbb{F}_p$, where a line is defined as a $1$-dimensional subspace. This is easy to see, if we draw ...
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1answer
25 views

Projecting a vector on orthogonal planes

I am looking from an engineer point of view. I have a sensor for which I need vector projecting on two different planes. I have the unit vector in the body frame that is to be projected and I obtained ...
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1answer
33 views

Isomorphic projective subvarieties, non-isomorphic rings

If $S \subset \mathbb{P}^n$ is a closed set (in the Zariski sense) then $\mathcal{I}(S) \subset k[x_0,\ldots,x_n]$ denotes the homogeneous ideal of polynomials which vanish at $S$. I want to find an ...
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Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
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36 views

A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically.

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically. A specific construction of a set of ...
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1answer
19 views

Trivial bundle on projective space?

On $P^n(\mathbb R)$ I consider the open sets $U_j$ (with $x_j \neq 0$) and the transiction functions of a linear vector bundle $E_d$, $f_{hk}:p\to(x_k /x_h)^d$. I have to demonstrate that, if $d$ is ...
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How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
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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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1answer
30 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
26 views

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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1answer
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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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1answer
36 views

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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1answer
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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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1answer
29 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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1answer
47 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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41 views

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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1answer
37 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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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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51 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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1answer
38 views

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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1answer
51 views

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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1answer
179 views

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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1answer
65 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 ...