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]

1464 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

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### 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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### 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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### 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

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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

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### 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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### 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

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### Give an explicit embedding from $\mathbb{R}P_2$ to $\mathbb{R}^4$

I have heard that the least dimension $m$ required for $\mathbb{R}P_2$ to be embedded in the Euclidean space is 4, thus I wanted to find an explicit formulae for it. I found two possible strategies, but is not sure that they'll work.
Define…

Michael Luo

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### 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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### Stiefel-Whitney numbers for product bundle

I'm reading Milnor's "characteristic classes" and I want to compute Stiefel-Whitney numbers of $ P^2 \times P^2 $ (product of projective spaces) for one of the problems,
I know how Stiefel-Whitney classes of a product bundle are related to the two…

Mehdi

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### Why is the line at infinity one-dimensional?

Why is the line at infinity a one-dimensional manifold, i.e. why is it truly a "line" at infinity and not a plane? Is my reasoning below at all correct?
(When I say "dimension" I mean "real dimension" if talking about $\mathbb{RP}^2$ and "complex…

Chill2Macht

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### Are all isomorphisms between the Fano plane and its dual of order two?

Famously from every finite projective plane $P$ we can create a dual projective plane $P'$ by taking the points of $P'$ to be the lines of $P$ and the lines of $P'$ to be the points of $P$.
It is also well known that the smallest projective plane,…

Vincent

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