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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37
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3answers
2k views

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 ...
16
votes
1answer
610 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 ...
13
votes
2answers
5k views

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 ...
11
votes
1answer
164 views

Homeomorphism $(\mathbb{CP}^1)^m/S_m \overset{\sim}{\to} \mathbb{CP}^m$?

For each $m \ge 1$, how do I construct a homeomorphism $(\mathbb{CP}^1)^m/S_m \overset{\sim}{\to} \mathbb{CP}^m$? My thoughts so far: I probably want to identify $\mathbb{C}^{m+1}$ with the vector ...
11
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1answer
209 views

$\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$

I have to solve the following: Show that $\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$ for $n\geq 2$. I have done this with knowledge of homotopy-groups, by showing that ...
10
votes
2answers
240 views

Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?)

This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering ...
9
votes
1answer
81 views

Two disjoint real projective planes in real projective space?

Let $\mathbb{R}\mathbb{P}^3$ be the real projective three-space. It is clear that any two hyperplanes in $\mathbb{R}\mathbb{P}^3$ intersect. But I wonder whether one could embed two copies of the real ...
8
votes
4answers
838 views

Orientability of $\mathbb{RP}^3$

I was wondering if there is a nice way to see that $\mathbb{RP}^{3}$ is orientable without using tools of algebraic topology, like homology. The only think I could think of was to argue that ...
8
votes
8answers
284 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 ...
8
votes
3answers
889 views

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, ...
8
votes
1answer
149 views

Line Bundle on subvarieties

I've been having problem actually restricting a Line bundle $L$ defined on some projective space $\mathbb C \mathbb P^{N-1}$ to a subvariety $X$. I know how to do this on an abstract level, but ...
8
votes
1answer
68 views

Parallelizing lines

Let $n \geq 1$ be an integer, and $L_1,\ldots,L_n$ be $n$ lines in $\mathbb{R}^3$ which are pairwise disjoint. Is it possible to move all $n$ lines continuously so that they never cross, and so as to ...
8
votes
1answer
210 views

What does projective space classify?

Let $A$ be a ring and let $\mathbb{P}^n = \operatorname{Proj} \mathbb{Z} [x_0, \ldots, x_n]$. Question. What does $\mathbb{P}^n$ classify? In other words, is there some kind of algebraic structure ...
8
votes
1answer
778 views

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 ...
7
votes
2answers
405 views

Coordinate ring in projective space. What are they?

When $X$ is an algebraic variety of affine $n$-space, then the coordinate ring of $X$ are polynomials restricted to $X$. But when $X$ is a variety of projective $n$ space, what are the elements ...
7
votes
2answers
136 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
7
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1answer
147 views

Trivial Restriction of Line Bundles

Say I have some projective space $\mathbb{P}^n$ and some line bundle $L=\mathcal{O}(-k)$. Now, I want to have a subvariety $Y$ in $\mathbb{P}^n$ such that $L\vert_Y$ is trivial. When is this the ...
7
votes
2answers
335 views

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 ...
7
votes
1answer
446 views

Cup Product Structure on the Projective Space

I am reading about cup products and am stuck on this exercise in Hatcher (3.2.5). Taking as given that $H^*(\mathbb{R}P^\infty,\mathbb{Z}_2)\simeq\mathbb{Z}_2[\alpha]$, how does one show ...
7
votes
1answer
171 views

Topology of the Segre product vs. the product topology

In general, the product topology on two (quasiprojective) varieties is not the same as the topology of the product variety given by the Segre embedding. This is something I've often seen asserted is ...
7
votes
1answer
115 views

Line bundles over $\mathbb R P^2$

As in this post, I'm continuing studying line bundles. Now it's line bundle over $\mathbb R P^2$. I know that this bundle is not trivial. So I want list up to equivalence all bundles over $\mathbb R ...
7
votes
0answers
149 views

Harris, Exercise 10.28 (weighted projective spaces)

So I recently started teaching myself about weighted projective spaces from Harris' Algebraic geometry. It was going well until I came across this exercise, which has me stumped: "Show that any ...
6
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2answers
169 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 ...
6
votes
1answer
177 views

Why is $0$ excluded in the definition of the projective space for a vector space?

For a vector space $V$, $P(V)$ is defined to be $(V \setminus \{0 \}) / \sim$, where two non-zero vectors $v_1, v_2$ in $V$ are equivalent if they differ by a non-zero scalar $λ$, i.e., $v_1 = ...
6
votes
1answer
208 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 ...
6
votes
1answer
624 views

Classifying Quasi-coherent Sheaves on Projective Schemes

I know some references where I can find this, but they seem tedious(Both Hartshorne and Ueno cover this). I am wondering if there is an elegant way to describe these. If this task is too difficult in ...
6
votes
1answer
127 views

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?
6
votes
1answer
518 views

rational functions on projective n space

How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
6
votes
1answer
178 views

Why is $\mathbb{R} P^n$ called projective space?

I know that: If one defines an equivalence relation on $\mathbb{R}^{n+1}-\{0\}$ by $$x\sim y \iff y=tx$$ for some nonzero real number $t$, where $x,y\in\mathbb{R}^{n+1}-\{0\}$, Then The real ...
6
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2answers
3k views

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 ...
6
votes
1answer
73 views

All tangential Stiefel-Whitney numbers of $\mathbb{R}P^q$ are zero iff $q$ is odd, reference?

Where can I find a reference to the proof of the following fact? All tangential Stiefel-Whitney numbers of $\mathbb{R}P^q$ are zero if and only if $q$ is odd. I made a quick search through ...
6
votes
1answer
216 views

Gap in Hartshorne I can't fill

Page 142, Example 6.11.4. I've been trying to go through the details of the sentence The proof of (6.10) shows that if $f \in K$ is invertible at $Z$, then the principal divisor $(f)$ on $X - Z$ ...
6
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1answer
386 views

Projective duality

Given a curve how do you intuitively construct the picture of its projective dual? I know points --> lines, lines--> points but for something like the swallowtail this is not really obvious.
6
votes
0answers
106 views

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = ...
6
votes
0answers
83 views

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 ...
6
votes
1answer
114 views

Cohomology of $\mathcal O(k)$

I am reading a paper in which it is claimed that $H^1(\mathcal O(-k),\mathcal O)=0$, where $k\geqslant 1$. Moreover, the argument also requires that $H^2(\mathcal O(-k),\mathcal O)=0$. Here ...
5
votes
2answers
127 views

Are homomorphisms into PGL related to the Schur multiplier?

I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field. I had been under ...
5
votes
3answers
438 views

Projective spaces with Zariski topology

Why $\mathbb{P}^1\times\mathbb{P}^1\not\cong\mathbb{P}^2$ where the projective spaces have the Zariski topology?
5
votes
1answer
155 views

Embedding of curves in projective spaces… typo?

I'm reading from the book "Geometry of algebraic curves", by Griffiths, Harris, Arbarello and Cornalba. In the middle of page 5 they define the map $\phi_{\mathscr{D}}:C\to \mathbb{P}V^*$, from a ...
5
votes
1answer
199 views

Proof you can't embed $\mathbb{P}^2$ into $\mathbb{R}^3$

I know you need to assume for a contradiction. The hard bit is the thing I can't do and that is show that if you do put $\mathbb{P}^2$ into $\mathbb{R}^3$ the path components of the complement are two ...
5
votes
1answer
44 views

Metric on real projective space

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 ...
5
votes
1answer
476 views

Ways of defining topology on $P^{n}(\mathbb{R})$

I was wondering if the following two ways of defining topology on $P^n(\mathbb{R})$ are the same and why? Since $P^n(\mathbb{R})$ is the quotient space of $\mathbb{R}^{n+1}$, define the topology on ...
5
votes
1answer
73 views

Can you define a vector space in terms of a pre-existing projective space?

Projective spaces are usually defined as the quotient of a vector space (by the equivalence relation that identifies collinear vectors). However, in my opinion, projective spaces seem intuitively less ...
5
votes
1answer
172 views

Cross-ratio relations

The way I define the cross-ratio in projectve geometry: Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
5
votes
0answers
20 views

Set of unordered $n$-tuples of points of $\mathbb{CP}^1$ can be identified with $\mathbb{CP}^n$? [duplicate]

This is a followup question to here. Fix $n \in \mathbb{N}$. What is the easiest way to see that the set of unordered $n$-tuples of points of $\mathbb{CP}^1$ can be identified with $\mathbb{CP}^n$?
5
votes
1answer
49 views

Write $\mathbb{P}^3_{\mathbb{C}}$ as a union of disjoint lines

Is there a set $\Gamma=\{L \subseteq \mathbb{P}^3_{\mathbb{C}}: L \textrm{ is a projective line}\}$ such that every point $p \in \mathbb{P}^3_{\mathbb{C}}$ lies on exactly one line $L_p \in \Gamma$? ...
5
votes
0answers
74 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 $\mathbb{C}^*.Id\cap SL_3 \cong C_3$ ...
4
votes
2answers
207 views

Which tangent bundles of real and complex projective spaces are trivial?

The tangent bundle of $S^m$ is known to be trivial only for $m=1,3,7$. Can this result be used to deduce that the tangent bundle of projective space $\mathbb{R}P^m$ is also trivial? (Clearly, for ...
4
votes
2answers
950 views

Finding zeros and poles of rational functions over elliptic curve

The matter of explicitly finding the order of a rational function on an elliptic curve in the projective plane at infinity (i.e. at the point $(0, 1, 0)$) still seems unclear. For example, Silverman ...
4
votes
3answers
260 views

what is the definition of a line in $\mathbb{P}^n(k)$ + how to compute the hilbert polynomial of two intersecting lines?

(1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic ...