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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“Projective” quotient of $\Bbb{Z}^2$

Consider the space of integer points $\Bbb{Z}^2=\{(x,y)|x,y\in\Bbb{Z}\}$. Consider now the equivalence relation: $$ (x,y) \sim (x',y') \quad \Leftrightarrow \quad \beta x'=\alpha x,\, \beta y'=\...
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Why is this isomophism of $PGL(2,\mathbb{Z})$ with a Coxeter group injective?

Let $W$ be a Coxeter group with generators $s_1,s_2,s_3$, where $m(s_1,s_2)=3,m(s_1,s_3)=2$, and $m(s_2,s_3)=\infty$. I understand that there's a surjective morphism $\varphi\colon W\to PGL(2,\...
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Existence of map between projective planes.

I'm struggling to solve this problem, any indications would be appreciated. Is there an application $f : \mathbb RP^3 \to \mathbb RP^1$ of class $\mathcal C^3$ such that $f^{-1}(p)$ is the union of ...
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How to find the transition function for two overlapping charts of $\mathbb{R}P^2$?

The real 2-dim projective space $\mathbb{R}P^2$ can be covered by the following 3 sets of unoriented lines through the origin un $\mathbb{R}^3$: $ U_x \doteq $ { all lines not lying in the yz plane} ...
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Isomorphisms and orthogonality in projective space.

I've been studying projective space in algebraic geometry for a few days from Perrin's 'Algebraic Geometry: An introduction'. In the first page of chapter III it reads ...projective space $P^n$ ...
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Do finite groups act admissibly on separated schemes of finite type over k

Background: Recall from SGAI that a group $G$ acts admissibly on a scheme $X$ if the quotient $X \to X/G$ exists and is an affine morphism of schemes. This is the case if and only if every orbit of $G$...
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Preimage of a simply closed curve under the two-dimensional antipodal map

Suppose $p:S^2\to P^2$ is the quotient antipodal map, and $J$ is a simply closed curve in $P^2$, then $p^{-1}(J)$ is either a simply closed curve in $S^2$, or two disjoint simply closed curves in $S^2$...
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21 views

Sum of projection vectors

Let say i have an orthogonal base $B$ which is some base of $R^3$, and now i want to find the vector coordinates of $[1,2,3]$ by base $B$ using the sum of projection vectors. 1.Is there some formula ...
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42 views

Radius of $\mathbb{CP}^n$

The question I am asking is basically Is it possible/usual to define a "radius" for certain metrics on $\mathbb{CP}^n$ in analogy to the case for $S^n$? To provide some more context about what I ...
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Definition of regular map from aquasi projective variety

These are from the book Basic Algebraic geometry by Shafarevich Definition of regular function on a quasi projective variety is as follows : Let $X\subset \mathbb{P}^n $ be a quasi projective ...
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QR factorization for least squares

This is from my textbook I don't undertand why small errorr in $A^TA$ can lead to large error in cofficient matrix? Because A=QR, so there should be no difference to use A or QR anyway.Could someone ...
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Verifying that the weighted projective space $\Bbb{P}_{\Bbb{Q}}(1,1,2,3)$ is singular.

I while ago I attended a talk that was somewhat over my head, and the speaker mentioned in passing that the weighted projective space $\Bbb{P}:=\Bbb{P}_{\Bbb{Q}}(1,1,2,3)$ is singular. I suppose this ...
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1answer
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cubic surface equation

If $[1,0,0,],[0,1,0],[0,0,1],[1,1,1],[1,3,2],[1,4,3]$ are six points on $P^2$ in general position and $f_0, f_1, f_2, f_3$ are the generators of the four dimensional vector space generated by cubics ...
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1answer
26 views

Geometric Significance that 2D Points Form a Line

I'm reading through Multiple View Geometry in Computer Vision, by Hartley and Zisserman, and on page 2 it is stated that points at infinity in the two-dimensional projective space form a line, and in ...
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1answer
43 views

Homotopy 'diagrams' for Klein bottle and projective plane

Background: I recently discovered that the complement to the circle and vertical axis shown below is homotopy equivalent to a torus Also complement to three infinite straight non-intersecting ...
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Associating to every line a vector in that line in an algebraic way

Let $X$ be a complex variety and $\mathscr E$ a locally free sheaf on $X$. Consider the fiber bundles $$ \mathbb P(\mathscr E) \overset{def}= \mathrm{Proj}(\mathrm{Sym}(\mathscr E)), \quad \mathbb A(\...
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28 views

Euler characteristic of the projective plane (using embedding diagram)

Make the square into the projective plane $\mathbb{P}$ by identifying edges and compute the Euler characteristic by embedding the following graph onto the surface: Here is my diagram of the ...
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35 views

Degree of maps $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$

In the book I am reading right now, it is defined that for a map $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$ the degree is the degree of the direct image cycle $\mu_{*}[\mathbb{P}^1]$. We are ...
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50 views

Is $\mathbb{R}P^1 \times \mathbb{R}P^1 = \mathbb{R}P^2$? If it isn't what does it look like?

We know that $\mathbb{R}P^1$ may be conceptualized as the set of all lines through the origin in $\mathbb{R}^2$. Alternatively, it may be conceptualized as a hemisphere of $S^1$ with antipodal points ...
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The subgroup of $PGL(V)$ stabilizing a projective configuration

Let $P(V)$ be a projective space and consider the natural action of $G=PGL(V)$ on it. Let $S=\{p_1,\dots, p_k\}$ be a finite set of points in $V$ where $k\geq 2$. Is there any reference about the ...
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1answer
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If two projective lines both intersect four given projective lines, must the two lines be parallel on an affine open?

L.S., I am trying to solve an exercise of my algebraic geometry course, which is as follows. Given four projective lines in $\mathbb{P}^3$, show that the number of lines intersecting them all is ...
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70 views

What would an infinite dimensional projective space look like as a scheme?

In topology, we can construct $\mathbb{CP}^\infty$ as the direct limit of $\cdots\rightarrow \mathbb{CP}^n \rightarrow \mathbb{CP}^{n+1}\rightarrow \cdots$ with the embedding given by $[x_0: x_1: x_2: ...
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Intersection of 3 or 4 dimensional subspaces of $\mathbb{P}^5$

What do we know about the intersection of 3 or 4 dimensional subspaces in $\mathbb{P}^5$? If we for example take two 4-dimensional subspaces of $\mathbb{P}^5$, what do we know about the dimension of ...
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An example calculated in Principles of Algebraic Geometry of Griffiths and Harris'.

On page 413 they write: Example. We can now make a second computation for Chern classes of projective space. Let $X_0, \ldots , X_n$ be linear coordinates on $\mathbb{C}^{n+1}$, and let $\mathfrak{...
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What properties single out $ \operatorname{Spec}(\mathbb{k}) $-schemes that are quasi-projective varieties over $ \mathbb{k} $?

I have a question in algebraic geometry that I would like to ask: Let $ \mathbb{k} $ be an algebraically closed field. Is there a property $ P $, phrased in the language of schemes, such that ...
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1answer
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Understanding problem 2.6 in Hartshornes algebraic geometry book .

Can anybody help me in understanding the hint given in the problem $2.6$ in Chapter $1$ of Hartshorne's Algebraic Geometry book ? I cannot see why $A(Y_i) $ can be identified with degree zero ...
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2answers
41 views

How to write the real projective plane as a pushout of a disk and the mobius strip?

I heard in topology class that the real projective plane is obtained by gluing a disk along the boundary of the mobius strip. I was wondering - how can I write this as a pushout? Also, how can I ...
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1answer
38 views

Projectivised tangent bundle of 2 sphere

I'm trying to understand how rotations act on the "projectivised" tangent bundle of the sphere. Let $S^2$ be the two sphere and denote by $P(TS^2)$ the tangent bundle where each tangent space $T_xS^...
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On Kapranov's “On the derived categories of coherent sheaves on some homogeneous spaces”

As a graduate master student I am reading Kapranov's paper "On the derived categories of coherent sheaves on some homogeneous spaces" (1988). One problem is that the paper assume lot of notations and ...
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Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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1answer
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tangent space of a curve in projective space

Suppose we have the curve $Z\subset \mathbb{P}^2$ given by the equation $y^2z-x^3=0.$ I have to find a basis for the tangent space at $(0:0:1)$, but I find the definition hard: Let $X$ be a variety ...
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What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...
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1answer
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Hilbert function and homogenous polynomials.

Let $\{[1:0:0],[0:1:0],[0:0:1],[1:1:1] \} = \{p_1,p_2,p_3,p_4\}$ be four points in the projective space $\mathbb{P}^2$. For every $p_i$, show there is a homogenous polynomial $f_i$ such that $$\left\{\...
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Constructing Incidence variety without using equations

Let $k$ be a field. Let $X$ be the Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a specified Hilbert polynomial. Let $Y$ be another Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a ...
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Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
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30 views

Using Riemann-Roch

I have canonical divisor $K$ of curve $\mathbb{P}^1$, and I would like to find $l(2K)$, $l(3K)$ and $l(-K)$ using Riemann-Roch theorem. I know that $g=0$ in this case, so $deg(K)=2\cdot 0-2=-2$. ...
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Hyperplanes without Axiom of Choice

For any projective space that contains more than one point, is it possible to prove that it contains a hyperplane without using the Axiom of Choice? It's easy enough to prove that there exists a ...
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Understanding functions in $\mathbb{P}^1$

I just need solution check: I am given function $f\in k(\mathbb{P}^1)$ that sends $(x:y) $ in $\frac{x}{y}$. If I understood this right, this is just coordinate function $x$ since it send and point $(...
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pencil of cubic curve passing six points

Let [1,0,0],[0,1,0],[0,0,1],[1,1,1],[1,3,2],[1,4,3] be a six points in general position. The question is how can determine the pencil of cubic curve passing through these points? Many thanks.
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projective nullstellensatz proof

I have to give a talk about projective varieties, including the projective nullstellensatz. As I'm not really into algebra or algebraic geometry, I've got some problems with the proof. Projectiv ...
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Group acting on a Projective Space

Let $G$ be an algebraic (zariski closed) subgroup of $SL(n,C)$ for some algebraically closed field $C$. Now $G$ acts on an $n$-dimensional vector space $V$ over $C$ where $V$ is a solution space of a ...
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1answer
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Projection on the subspace perpendicular to a vector

I am running an algorithm and during one of the steps of the algorithm, I have to update a matrix $B$ but projecting every column of the matrix, $B_j$, j=1,...,k, on the subspace perpendicular to a ...
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1answer
77 views

Lines on a Quintic Threefold

We work over an algebraically closed field $k$. I've been given the exercise of showing (using only the technology introduced in the first chapter of Shafarevich's Basic Algebraic Geometry) that ...
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Rings of Regular functions, and regular maps between Quasi Affine to Quasi Proj. Varieties.

I have studied classical algebraic geometry a while ago. I want to sum up in short as possible everything regarding their rings of regular functions. If my understanding not correct, please correct me....
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describing proj. seurface.

I have the surface $W=Z(x_0x_1-x_2x_3)$, in $\mathbb{P^3}$ and I want to describe it as a union of an affine piece and some other piece laying in $\mathbb{P^2}$. My solution is to look at: 1-$W\...
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Intersection of circle and line.

Now we can realize that intersection of two parallel lines at point of infinity in projective space. I can manage to visualize this and compute it. My issue is if we have the two varieties $ X_1= Z ...
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1answer
58 views

Show that any two distinct lines in $\Bbb P^2$ intersect in one point.

Show that any two distinct lines in $\Bbb P^2$ intersect in one point. Proof(My attempt). Let $L_1, L_2$ be any two distinct lines in $P^2$. Write $L_i = V (a_iX + b_iY + c_iZ), i = 1,2$. It ...
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Localization of the ideal of some projective variety

Let $X \subset P^n$ be some projective variety, and $I$ be its ideal. We consider the 0th graded piece of the localization $(I \cdot k[Z_0,...,Z_n,Z_i^{-1}])_0$ in the coordinate ring $(k[Z_0,...,...
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1answer
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All translations on $\mathbb{R}$ are induced by a $2\times2$-matrix

I want to prove that all translations on an affine straight line are induced by projective transformations of the projective extension. If we look at $\mathbb{R}$ as our affine straight line, then ...
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Examples of homogeneous polynomials that define $\mathbb{P}^n$

The complex projective space $\mathbb{P}^n$ is also a projective variety. According to Hartshorne, a projective variety is defined a s the zero set of a subset of homogeneous polynomials defined on $\...