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1answer
16 views

Mipoint between points in projective space

Is there a way to define the midpoint between points in projective space?
2
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2answers
48 views

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 ...
2
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1answer
33 views

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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0answers
13 views

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 ...
0
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1answer
18 views

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 ...
4
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3answers
67 views

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 $\mathbb R^{n+1} \setminus \{0\}$ ...
0
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1answer
53 views

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 ...
2
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1answer
43 views

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 ...
0
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1answer
21 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 ...
-1
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1answer
30 views

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 ...
2
votes
1answer
38 views

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 ...
2
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0answers
30 views

General form of regular maps $\mathbb{P}^n\to\mathbb{P}^m$?

I'm reading through Milne's algebraic geometry notes, and there's a remark without justification I'm having trouble seeing. Essentially: Suppose $F : \mathbb{P}^n\to\mathbb{P}^m$ is a regular map ...
1
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1answer
46 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 ...
1
vote
1answer
70 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 ...
10
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1answer
195 views

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 ...
1
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0answers
17 views

Complex projective space and its dual are homeomorphic?

Consider $\mathbb{C} P^n$ and its dual space, which consists of hyperplanes in $\mathbb{C} P^n$. Are they homeomoprhic? I read this fact somewhere, but can't remember where. Also i don't even ...
1
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2answers
18 views

Why is the projection of a vector V onto a span W, independent of the orthogonal basis of W.

Very straightforward question. I have read time and again in my book that it is independent but I don't understand why? Wouldn't changing the basis mean changing the length of the projection?
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0answers
20 views

Why are closed discs preserved by linear fractional transformations in non-archimedean geometry?

If $K$ is a local field equipped with a non-archimedean metric, then I have read in several places that the action of $PGL_2(K)$ by linear fractional transformations takes closed discs to closed ...
0
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1answer
107 views

Is the complex projective plane a compact manifold with or without boundary (closed manifold)?

my question is the one in the title. (My motivation is to understand in which way Freedman's classification of compact simply-connected 4-manifolds implies the Poincare conjecture for 4-manifolds, as ...
1
vote
1answer
45 views

Reference request: second Chern class of P^2

I have heard that $c_2(T_{\mathbb{P}^2})=e(\mathbb{P}^2)$. What's the general result and where can I read about it? Thanks.
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0answers
62 views

How to prove that $Aut(\mathbb{P}^2) \cong PSL_3 (\mathbb{C})$?

Notation: $\mathbb{P}^2$ denotes complex projective plane. We have an action $$SL_3 \times \mathbb{P}^2 \to \mathbb{P}^2, \ (A,[v])\mapsto [Av]$$ with kernel $\mathbb{C}^*.Id\cap SL_3 \cong C_3$ ...
0
votes
1answer
47 views

A birational map from $\mathbb{P}^1$ to an irreducible plane projective curve

Let $C$ be an irreducible plane projective curve described by the equation $$zf(x, y) + g(x, y) = 0,$$ where $f$ and $g$ are a homogenous forms of degree $d - 1$ and $d$, respectively. What would be ...
1
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1answer
18 views

projective space over finite fields

Let $A,B$ be sets non empty sets. Let say that if $p\in A$ then $p$ is said to be a point and if $l \in B$ then $l$ is said to be a line. Let $C$ be a set of the form $\{p,l \}$ with $p \in A$ and ...
2
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1answer
31 views

Comparison between two definitions of real projective spaces.

The most common definitions of real projective spaces are: $\mathbb{R} \mathbb{P} ^n = (\mathbb{R}^{n+1} - 0)/ \sim$, where $x,y \in \mathbb{R}^{n+1}-0$ satisfies $x \sim y$ iff $x = \lambda y$ for ...
2
votes
1answer
55 views

Axioms vs Models in projective and hyperbolic geometry

I'm studying Projective Geometry. The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. ...
0
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0answers
61 views

Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows ...
1
vote
1answer
23 views

Unitary transformation of Fubini-Study metric

I am trying to solve a problem in Introduction to Complex Geometry by D. Huybrechts, question 3.1.6 which is the following: let $A\in GL(n+1, \mathbb{C})$ be a $\mathbb{C}$-linear transformation ...
1
vote
1answer
38 views

Cohomology of intersection of hyperplanes

let $X = H_1 \cap ... \cap H_d$ be a compact submanifold of $\mathbb{P}_N$ where the $H_i$ are hyperplanes. I want to compute $H^q(X, \mathcal{O}_{\mathbb{P}_N}(m)|X)$. I am pretty unexperienced in ...
1
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0answers
49 views

Exceptional coherent sheaves on $\mathbb{P}^n$ are vector bundles

Let $E$ be a coherent sheaf over $\mathbb{P}^n_k$. Coherent sheaf is called exceptional if $\operatorname{Hom}(E,E) \cong k$, and $\operatorname{Ext}^{> 0}(E,E) \cong 0$. How one can show that such ...
1
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1answer
33 views

Help on formalisation proof of the triviality of a kernel in Mayer-Vietoris

Consider the Mayer-Vietoris sequence for $\mathbb{RP}^2$, where the two open sets are $U:= \{ [x;y;z] \in \mathbb{RP}^2 | z \neq 0 \}$ and $V = \mathbb{RP}^2 \setminus [0;0;1]$. I've proved that $U ...
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8answers
141 views

How to show $P^1\times P^1$ (as projective variety by Segre embedding)is not isomorphic to $P^2$?

I am a biginner. This is an excise from Hartshorne Ch 1, 4.5. By his hint, it seems this can be argued that there are two curves in image of Segre embedding that do not intersect with each other ...
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0answers
19 views

Embedding into the projective space - tangent level injection

I am reading Griffiths and Harris, the section of embedding a manifold $M$ into projective space. Let $\mathcal{L}$ be a line bundle over $M$, with dim $H^0(M,\mathcal{L})=N$, and let $s_0,...,s_N$ be ...
4
votes
2answers
105 views

Find diagonal of inverse matrix

I have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the cholesky ...
3
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2answers
99 views

Is the Projective Real Plane Compact?

I feel like $\Bbb P (\Bbb R^2)$ is compact, but I know that $\Bbb R^2$ is locally compact, therefore it has a one-point compactification. $\Bbb P (\Bbb R^2)$ adds more than one point to the real ...
0
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1answer
41 views

$\mathbb{R}P^2$ and its fundamental group by identification of edges of unity square

Suppose we identify edges of the the unity square $[0, 1] \times [0, 1]$, as in the picture: http://de.wikipedia.org/wiki/Datei:ProjectivePlaneAsSquare.svg Now to compute the fundamental group, for ...
1
vote
1answer
24 views

The projection operator defined by $(P_n(h) - h, v)_H = 0$

Let $V \subset H$ be separable Hilbert spaces with continuous embedding and suppose $\{v_n\}$ be a (non-orthogonal) basis for $V$. If we let $V_n = \text{span}(v_1, ..., v_n)$ and given $h \in H$ we ...
1
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2answers
62 views

Parallel Lines Intersecting in the Projective Plane

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that ...
0
votes
1answer
43 views

how to compute the distance between a matrix and its lower rank approximation?

I have a matrix $X$ and $Z$ a lower rank approximation of $X$ obtained using only few of the columns of $X$. I would like to have a measure of how distant are $X$ and $Z$. In particular I would like ...
0
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0answers
30 views

Normalize matrix so that its projection equals identity - which to use?

I wish to normalize a given matrix M, n by k, so that its projection matrix equals the identity. This is to speed up computations. The transformation/decomposition would look like this: Generate M by ...
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0answers
19 views

Saari's homographic conjecture and the actual definition of homography

By the wikipedia definitions found here and here, and especially by the definition implicit in this MSE post, it seems two images are homographic if they are renderings of the same set of points in ...
1
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0answers
18 views

Measures on $\mathbb{P}^1(\mathbb{C})$ and $\mathbb{P}^1(\mathbb{C}_p)$?

Is there a natural measure on the above spaces? Ideally, I would like a measure that is invariant under automorphisms of $\mathbb{P}^1$.
4
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1answer
150 views

Blow Up: Resolution of Singularity

For blow ups, I have worked only in $\mathbb{CP}^2$. Once I locate the base-point, say $[x,y,z]=[0,1,0]$, I go back to $\mathbb{C}^2$ by considering the chart $y=1$. I then proceed to blow up ...
1
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1answer
70 views

Some questions about complex curves in $\mathbb CP^2$

I would like to ask for some clarifications in the following questions about complex curves. My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a ...
1
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0answers
33 views

Morphisms of quasi-projective varieties

Let $Y\subseteq \mathbb{P}^n(k)$ be a quasi-projective variety. By Görtz, Wedhorn (page 32, Proposition 1.65) in order to show that $$h:Y\to \mathbb{P}^m(k), y\mapsto (f_0(y):\dots :f_m(y))$$ is a ...
2
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1answer
102 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
2
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2answers
106 views

Projective varieties and irreducibility

The "modern"(schematic) definition of a projective variety is the following: Let $k$ be an algebraically closed field. A projective variety over $k$ is a closed subscheme of $\mathbb ...
1
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1answer
38 views

What is the line going through points $(5, 5, 5), (2, 2, 2) \in \mathbb{R^3}$ when mapped it is mapped to a point in the real projective plane?

So the real projective plane is homeomorphic under a function $f$ to $\mathbb{R^3} - (0, 0 ,0)$. Hence lines in $\mathbb{R^3} - (0, 0 ,0)$ become points in the the real projective plane. So what is ...
6
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1answer
181 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
0
votes
1answer
37 views

Isomorphism between projective space and grassmanian $\mathrm{Grass}(1,n+1)$

I have to show that the projective space $\mathbb{P}^n$ is isomorphic to the grassmanian $\mathrm{Grass}(1,n+1)=\{V\subseteq\mathbb{R}^{n+1}:V\,\text{linear subspace,}\,\dim\,V\,=1\}$ as well as ...
2
votes
1answer
19 views

Vector space and its Projecctivized Space

Why is the co-dimension one subspaces are the points of $\mathbb P(V^{\vee})$. $V^{\vee}$ is the dual space of V and and $\mathbb P(V)$ is the projectivized space of V. $\mathbb P(V)= ...