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Questions tagged [projective-space]

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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Uniqueness of the decomposition of rank 2 holomorphic line bundle over $\mathbb{CP}^1$

Let $\mathcal{O}(1)$ be the hyperplane bundle over $\mathbb{CP}^1$ and $\mathcal{O}(n)=\mathcal{O}(1)^{\otimes n}$. I knew that any rank 2 holomorphic line bundle over $\mathbb{CP}^1$ is isomorphic to ...
Rosalina's user avatar
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Čech cohomolog of a good cover of the real projective plane

I was reading the Bott/Tu book and trying compute the Čech cohomology of a good cover of the real projective plane (exercise 9.10). The nerve of the cover is dipicted as below: we glue the opposite ...
Xipan Xiao's user avatar
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Kahler-Einstein metric on complex projective space

I think this question may be well-known to the experts; or someone may have already asked the following question in this website. Since I couldn't figure it out myself and I couldn't find a related ...
Tong's user avatar
  • 139
2 votes
1 answer
98 views

Topology of the projective space

I am studying the fundamental group and I am facing a problem with the projective space: we know that, by definition, the real projective space $P^m(\mathbb{R})$ is the quotient of $R^{m+1}-\{0\}$ ...
user1255055's user avatar
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Is this Correct about the Points in the Projective Space?

For each $i=0,…,n$, let $U_i=\{(x_0:…:x_n)\in \mathbb{P}^n(k)|x_i\neq 0\}$. Given $U_i\subseteq \mathbb{P}^n(k)$, I want to identify the points that are in the subsets $U_2$, $U_2\cap U_3$ and $\...
Mr Prof's user avatar
  • 451
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1 answer
61 views

Cross-ratio for more than 4 points on a line

It is known that the pairs $(\mathbb{P}^1,4 \mbox{ points})$ are classified by a $1$-dimensional family, parametrized by cross-ratio (up to an action of $S_3$). I would like to ask if the same is true ...
L_b's user avatar
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Stereographic Projection Using a One-Sheeted Hyperboloid

This is in reference to the stereographic projection of a one-sheeted hyperboloid, as detailed on page 199 of this book. The author visualises the inversive Minkowskian plane by using a stereographic ...
Anomander Rake's user avatar
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60 views

Explicit linear system corresponding to rational map to $\mathbb{P}^1$

Let $L$ be a line in $\mathbb{P}^3$. Then we can define a map $$ \pi\colon \mathbb{P}^3 \dashrightarrow \mathbb{P}^1, \qquad x \mapsto \langle x, L \rangle \cap L' $$ where $L'$ is a line such that $...
fish_monster's user avatar
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Quadric equation in Projective Geometry

It is well known that 5 points in $\mathbb{R}^2$ define a conic. While studying the book Geometry II by Marcel Berger, I came across a theorem stating that 9 points in $P\mathbb{R}^3$ define a quadric....
User0212's user avatar
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Showing the Graph of a Projective Transformation is Intersection of a Quadric Surface and a Plane

I'm trying to do Exercise 3.6 at the end of this pdf: Let $\tau: P^1(\mathbf{R}) \rightarrow P^1(\mathbf{R})$ be a projective transformation and consider its graph $$ \Gamma_\tau \subset P^1(\mathbf{...
hbghlyj's user avatar
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Difference between $PGL(2,\mathbb{Q}_p)$ and $GL(2,\mathbb{Q}_p)$ in terms of determinant

I'm trying to understand the difference between $PGL(2,\mathbb{Q}_p)$ and $GL(2,\mathbb{Q}_p)$ in terms of determinant. For example, in the real case, $PGL(2,\mathbb{R})$ is $GL(2,\mathbb{R})$ up to ...
The way of life's user avatar
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2 answers
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Defining the real projective line via equivalence relation

I'd just like to know if my approach is correct. Let $\sim$ be an equivalence relation on $\mathbb R^2$ defined as: $$(x_1,y_1)\sim(x_2,y_2) \iff \exists\lambda\in\mathbb R\smallsetminus\{0\} : x_1 = \...
Elvis's user avatar
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What's the point of the local zeta function?

I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see ...
Samuel Johnston's user avatar
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Finite Order (in Homology Group) Implies Non-Orientability? (Intuition)

The motivation for this question comes from this post ("Where does the term "torsion" in algebra come from?"). To keep it brief: Given an element of finite order $\in H_n(X)$ (...
JAG131's user avatar
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Which representation of $\mathrm{SL} (2; \mathbb{C})$ does the tautological bundle on $ \mathbb{CP}^1 $ correspond to?

If I understand correctly, by Borel-Weil-Bott, for each weight of $\mathrm{SL} (2; \mathbb{C})$ there is a corresponding (holomorphic) line bundle / invertible sheaf on $ \mathbb{CP}^1 $. Denoting the ...
smitke6's user avatar
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Intersection multiplicity of two curves embedded in the projective plane

I encountered the following problem and hope to receive guidance from everyone. Let $\phi$ maping from $C[x,y]$ into $C[x,y,z]$ turning a polynomial $f(x,y)$ into $F(x,y,z)= z^{deg f}f(x/z,y/z)$. ...
ChaosLord's user avatar
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31 views

Inverse of pullback metric

Suppose we have the inclusion $\iota: X \hookrightarrow \mathbb{P}^n$ which is injective, but not a diffeomorphism. Given the standard metric $g_{\mu \overline{\nu}} dz^{\mu} \otimes d\overline{z}^{\...
Eweler's user avatar
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1 answer
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The line bundle $\mathcal{O}(1)$ over $\mathbf{CP}^{1}$

$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}\newcommand{\Sheaf}{\mathcal{O}}$Note to posterity: This question got asked and deleted. As suggested in meta I'm reposting and including ...
Andrew D. Hwang's user avatar
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1 answer
42 views

Find the equation of the closure of a curve in the projective space [closed]

Let k be an algebrically closed field and consider $C \subset \mathbb{A}^2(k)$, the curve of equation $f(X,Y)=0$ with $f(X,Y)=X(X-1)(X+1)-Y^2$. We want to find the equation of its closure in $\...
BillyJohny's user avatar
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Menelaus $\implies$ Intercept theorem?

Here is a proposed algebraic demonstration of Menelaus's Theorem Let $A, B, C\in \mathbb R^3\setminus \{0\}$ a basis of $(\mathbb R^3,+,.)$. In this basis, $$A=1.A+0.B+0.C; B=0.A+1.B+0.C; C=0.A+0.B+1....
Stéphane Jaouen's user avatar
1 vote
1 answer
64 views

Chern class of $\mathcal O(d)$

I was reading Vincent Bouchard notes notes. On section $3.3.2$, I face some issue: Let $X$ be a smooth hypersurface in $\mathbb{C P}^m$ defined as the zero-locus of a degree $d$ polynomial $p$. We ...
N00BMaster's user avatar
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1 answer
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Exercise regarding projective space $\mathbb{C}P^{n-1}$

Consider the set $\mathbb{C}^n \setminus{0}$ and define the following equivalence relation on it: $x \sim y$ if there exists $z \in \mathbb{C}$ such that $y=zx$. Let $\mathbb{C}P^{n-1}$ denote the ...
Philip's user avatar
  • 635
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1 answer
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What's the relationship between the sections of $\mathcal O(1)$ and $T^{(1,0)} \mathbb{C P}^m$?

I was reading Vincent Bouchard notes. On section $3.3.1$, I face some issue: First, we need to compute the total Chern class of $\mathbb{C P}^m$. We recall that homogeneous coordinates $z_i, i=0, \...
N00BMaster's user avatar
1 vote
1 answer
45 views

Examples of homogeneous ideals with $J \cap B_+ \subset \sqrt{I}$ but $J \not\subset \sqrt{I}$.

This is a question about a Lemma in Liu's Algebraic Geometry and Arithmetic Curves. The result states the following (this is Lemma 2.3.35, paraphrased to focus only on the part I am asking about): ...
stillconfused's user avatar
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1 answer
66 views

How to visualize different line bundle over complex projective space $\mathbb{CP}^1$

We knew that the projective space $\mathbb P^1$ is the $1$-dim subspace of $\mathbb A^2$ (affine space). From Fig 1 and 2 we have, \begin{align} \phi_x:\mathbb A^1_x&\rightarrow\mathbb P^1\\ x&...
Nahar's user avatar
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0 answers
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Proving that $ \{([w_0 : w_1], [z_0 : z_1 : z_2]) \in \Bbb {CP}^1 \times \Bbb {CP}^2| w_0z_1 = w_1z_0\}$ is a smooth complex hypersurface

Let F be the subset of $\Bbb {CP}^1 \times \Bbb {CP}^2$ defined by $F = \{([w_0 : w_1], [z_0 : z_1 : z_2]) \in \Bbb {CP}^1 \times \Bbb {CP}^2| w_0z_1 = w_1z_0\}$ I want to prove that F is a smooth ...
some_math_guy's user avatar
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0 answers
53 views

Pullback of vector field under inclusion map

Suppose we have a manifold $X$ embedded in ambient space (this is $\mathbb{P}^n$), $$ \iota: X \hookrightarrow \mathbb{P}^n. $$ Given a vector field on $\mathbb{P}^n$, is there any way, canonical or ...
Eweler's user avatar
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1 answer
70 views

Is it okay to explain Projective line with lines through origin in $\mathbb R^2$?

I want to start with definitions. Affine space $\mathbb A^n$ over the field $K$ is the set of $x_i$'s i.e., $\mathbb{A}^n = \{(x_1,\dots,x_n):x_i \in K \}$ if $n=2$, projective line defined to be $\...
Elise9's user avatar
  • 193
2 votes
0 answers
56 views

Projective curve covered by two affine pieces

A non-singular projective curve $X$ is covered by two affine pieces (with respect to different embeddings) which are affine plane curves with equations $y^2 = f (x)$ and $v^2 = g(u)$ respectively, ...
ForgeBloyb's user avatar
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0 answers
30 views

Determine if point can be projected on any segment in polyline

I am trying to implement a script that determines if a series of points can be projected on any segment of a polyline. The projection must be exactly on the segment, not in the rest of the line ...
NeuroTheGreat's user avatar
4 votes
2 answers
131 views

Showing that $x/y$, where $x=x_1/x_0$ and $y=x_2/x_0$, is a local parameter of $x_0x_2^2=x_1(x_1-x_0)(x_1-2x_0)$ at $p=(0:0:1)$

This is question 5 from https://www.dpmms.cam.ac.uk/study/II/AlgebraicGeometry/2023-2024/HW4.pdf. Consider the curve $$ x_0x_2^2 = x_1(x_1-x_0)(x_1-2x_0)$$ over a field of characteristic zero. I need ...
ForgeBloyb's user avatar
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8 views

Central projection of $\mathbb{E}_3^{\star}$

Let $\mathbb{E}_3^{\star}$ be the extended euclidean space. Given a plane $\alpha$ and a central projection $\psi:\mathbb{E}_3^{\star}\backslash\{S\}\to\pi$, how do I prove that $S\in\alpha\...
XIC22's user avatar
  • 399
2 votes
0 answers
38 views

Definition of projective space

I am reading Silverman's 'The Arithmetic of Elliptic Curves'. It says $\mathbb{P}^n(K)=\{[x_0,\cdots,x_n]\in\mathbb{P}^n:\text{ all }x_i\in K\}$, and then a remark says: if $P=[x_0,\cdots,x_n]\in\...
Eulerian's user avatar
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32 views

$AG(2,2)$ and $AG(3,2)$ and their resp. triangular (Fano plane-like) and tetrahedral ($PG(3,2)$-like) representations. Are my assumptions correct?

From the "Finite affine planes" section of the Wikipedia article "Affine plane (incidence geometry)", in a (finite) affine plane of order $n$: each line contains $n$ points, each ...
Kevin M. Lamoreau's user avatar
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1 answer
67 views

How can I show that the quotient function to the real projective space is closed?

As I have to demonstrate that this space is a Hausdorff space, obviously I hope to find a proof without the use of this fact
George's user avatar
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1 answer
255 views

Global sections of the line bundle $\mathcal{O}(k)$, Prop. 2.4.1 Huybrecht's $\mathbb{C}[z_0,\ldots,z_n]_k\cong H^0(\mathbb{P}^n,\mathcal{O}(k))$

We need to show that the map $$\theta:\mathbb{C}[z_0,\ldots,z_n]_k\to H^0(\mathbb{P}^n,\mathcal{O}(k)) $$ Is an isomorphism. Huybrecht's describes the map $\theta$ right before the proposition. This ...
領域展開's user avatar
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1 answer
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Difference between the sphere and RP^2 in the generators and relations of $\pi_1$

Here are two drawings we had that I did not quite well understand why they are correct: Here are my questions: 1-I think the drawing of $RP^2$ is meant to be a circle not a disk, am I correct? 2- ...
Intuition's user avatar
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0 answers
37 views

Set is equal to its closure in projective space [duplicate]

I'm trying to determine some criteria for when the closure of a closed irreducible set $X$ of $\mathbb A^n$ is equal to the set itself. That is if $\overline{X}$ is the closure of $X$ in $\mathbb P^n$ ...
Jeff's user avatar
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0 answers
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Irreducible components of an affine cone are affine cones [duplicate]

We have shown that for an affine cone of a nonempty subset $X$ of $\mathbb P^n$, i.e $C(X)$, that the irreducible decomposition of $C(X)$ given by $Y_1 \cup \dots \cup Y_m$ has that $Y_i = C(X_i)$ for ...
Jeff's user avatar
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0 answers
53 views

How to think about projective geometry problems

This semester I am taking an introductory course on projective geometry. Even though I believe I understand the different theoretical concepts correctly (supplementary varieties, Grassmann Formula, ...
tsch_'s user avatar
  • 31
0 votes
1 answer
59 views

If $\phi:V_1\rightarrow V_2$ is a morphism of varieties then $V_1\cong \phi(V_1)$

I am reading Silverman's The Arithmetic of Elliptic Curves. I am wondering if with the definition of morphism he gives, we can conclude that if $\phi:V_1\rightarrow V_2$ is a morphism of projective ...
kubo's user avatar
  • 2,067
1 vote
0 answers
90 views

Why $\mathbb{CP}^1 \cong S^2$? [closed]

Consider $\mathbb{CP}^1 =\frac{\mathbb{C}^2\setminus \{ 0\}}{\mathbb{C}^*}$, one dimensional projective Hilbert space. I was wondering if someone could help me about proving $\mathbb{CP}^1 \cong S^2$ ...
Mahtab's user avatar
  • 751
2 votes
0 answers
50 views

Isotropy subgroup of $\operatorname{GL}_{n+1}(\mathbb R)$ acting on $\mathbb R \mathbb P^n$

The general linear group $\operatorname{GL}_{n+1}(\mathbb R)$ (as a Lie group) acts smoothly on the real projective space $\mathbb R \mathbb P^n$, via $A \cdot [x] := [Ax]$. By $[x]$ here, I mean the $...
Joseph Kwong's user avatar
0 votes
1 answer
83 views

Action of $SL_2(\mathbb{Z})$ on the projective plane over $\mathbb{Z}_p$

The group $SL_2(\mathbb{Z})$ act on the projective spaces $P(\mathbb{Z}_p)$ and the upper half of the complex plane $\mathbb{H}$ by linear fractional transformations. I am wondering whether there is a ...
QMath's user avatar
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2 votes
1 answer
80 views

Can space be augmented with a plane at infinity so that parallel planes intersect at a line at infinity?

The real plane can be augmented with a line at infinity such that two parallel lines intersect at a point at infinity, and the set of all such points forms a line. In space (3 dimensional solid ...
SRobertJames's user avatar
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1 vote
0 answers
66 views

Topology of blow up of a cone

Let $X$ be a real algebraic surface with a unique $A_1$ point at $p \in X$ (that is, the germ of $X$ at $p$ is isomorphic to the germ of $\{x^2+y^2=z^2\}$ at the origin). Let $Y \to X$ be the blow up ...
Serge the Toaster's user avatar
3 votes
1 answer
201 views

Example 3.5 Silverman's Elliptic Curves: problem with a point in the projective space

I am reading Silverman's The Arithmetic of Elliptic Curves. I have a question regarding Example 3.5, which is the following: I have doubts about the equality $[-Y^2,Y(X-Z))]=[-Y,X-Z]$. Two points $[...
kubo's user avatar
  • 2,067
0 votes
1 answer
51 views

Parallel lines diverge behind the observer in projective geometry

Of course parallel lines converge at a point at infinity in projective geometry, but visually, they appear to diverge as one gets closer and closer to the start of one's vision, i.e. they diverge ...
Svenn's user avatar
  • 51
2 votes
2 answers
147 views

Tangent space of $\mathbb{P}(V)$

I encountered this problem studying the local period map and I'm wondering how to solve it. I would like to prove that, given $V$ a complex vector space and $W \subseteq V$ a one-dimensional subspace, ...
WindUpBird's user avatar
1 vote
0 answers
36 views

Analogue of complex projective space, replacing GL1 with GLn

$\mathbb{CP}^1$ can be formed from $\mathbb{C}^\times = \text{GL}_1$ by gluing $\mathbb{C}$ by itself along $\mathbb{C}^\times$, a pushout of $1/z,z : \mathbb{C}^\times \rightarrow \mathbb{C}$. I am ...
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