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
Questions tagged [projective-space]
1758 questions
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Why the emphasis on Projective Space in Algebraic Geometry?
I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed.
Why does Miranda (and from what little I've seen, algebraic geometers in general) place…
Potato
- 41,411
62
votes
2 answers
Lines in projective space
I have the following definitions:
Given a vector space $V$ over a field $k$, we can define the projective space $\mathbb P V = (V \backslash \{0\}) / \sim $ where $\sim$ identifies all points that lie on the same line through the origin.
A…
Jonathan
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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 respectively. 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 curious, is it possible to find a…
Clara
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27
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Orientability of projective space
Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even.
First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with the Möbius strip.
A n-dimensional manifold is…
bbnkttp
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What is the difference between projective geometry and affine geometry?
I recently started reading the book Multiple View Geometry by Hartley and Zisserman. In the first chapter, I came across the following concepts.
Projective geometry is an extension of Euclidean geometry with two lines always meeting at a point.
In…
rotating_image
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20
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9 answers
How to show $P^1\times P^1$ (as projective variety by Segre embedding) is not isomorphic to $P^2$?
I am a beginner.
This is an exercise from Hartshorne Chapter 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 while in $P^2$ any two curves intersect.
I feel…
user48537
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How is the metric defined on the real projective space $\mathbb{RP}^n$?
The standard metric on $RP^n$ is usually defined to be the metric that locally looks like the metric on $S^n$. But as a differentiable manifold (and not just as a set), $RP^n$ is not a subset of $S^n$, it is a quotient. So there is no natural map…
geodude
- 8,357
19
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1 answer
Homotopy groups of $\mathbb{RP}^\infty$, $\mathbb{CP}^\infty$.
Could someone supply me a precise reference to the computation of all homotopy groups of infinite real projective space and infinite complex projective space?
user203482
17
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2 answers
Why are there no non-trivial regular maps $\mathbb{P}^n \to \mathbb{P}^m$ when $n > m$?
Question. Let $k$ be an algebraically closed field, an let $\mathbb{P}^n$ be projective $n$-space over $k$. Why is it true that every regular map $\mathbb{P}^n \to \mathbb{P}^m$ is constant, when $n > m$?
I can't see any obvious obstructions: there…
Zhen Lin
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Computing the Fubini-Study metric in affine chart
Someone asked this question recently and then deleted it, but I still would like to figure out the answer.
Let us try to compute the Fubini-Study metric in inhomogeneous coordinates on $\mathbb{C} P^n$, using the Hopf fibration. For simplicity, let…
Seub
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?
I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, and that there is a way to generalize…
pregunton
- 6,358
16
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2 answers
How to define a Riemannian metric in the projective space such that the quotient projection is a local isometry?
Let $A: \mathbb{S}^n \rightarrow \mathbb{S}^n$ be the antipode map ($A(p)=-p$) it is easy to see that $A$ is a isometry, how to use this fact to induce a riemannian metric in the projective space such that the quotient projection $ \pi:…
Lonely Penguin
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15
votes
4 answers
Find diagonal of inverse matrix
I have computed the Cholesky of a positive semidefinite 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 that I have computed? Or will finding the…
sachinruk
- 1,001
15
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1 answer
Tangent bundle of a quotient by a proper action
Given a compact group $G$ acting freely on a manifold $X$, is there a "nice" way to describe the tangent bundle of the quotient $X/G$ (when it is a manifold)?
In the case the group $G$ is finite, or more generally when its action is properly…
Najib Idrissi
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15
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3 answers
Formula for ellipse formed from projecting a tilted circle onto the xy plane
A circle is drawn in the center of a piece of paper. The piece of paper is placed on a table and a camera is positioned directly overtop to look at the circle.
A piece of glass is placed between the camera and the piece of paper and is parallel to…
denversc
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