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30
votes
3answers
1k 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 ...
11
votes
1answer
194 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
1answer
184 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 ...
10
votes
2answers
3k 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 ...
9
votes
2answers
230 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 ...
8
votes
4answers
718 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
140 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
607 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
134 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
204 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 ...
7
votes
2answers
267 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
1answer
97 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 ...
6
votes
1answer
165 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
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 ...
6
votes
1answer
117 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 ...
6
votes
1answer
468 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
247 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 ...
6
votes
1answer
142 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
votes
1answer
268 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.
5
votes
2answers
102 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
356 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
190 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
2answers
254 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 ...
5
votes
1answer
259 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 ...
5
votes
1answer
347 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
62 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
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 $\mathbb{C}^*.Id\cap SL_3 \cong C_3$ ...
5
votes
1answer
95 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
0answers
459 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 ...
4
votes
3answers
218 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 ...
4
votes
2answers
270 views

is there a decomposition of $L_2(q)$ into a direct product?

I wonder if there is a nontrivial decomposition of the $L_2(q)$, where $q$ is a prime power, into a direct product. I think that there is none, but I am not sure. $L_2(q)$ refers to the special ...
4
votes
1answer
86 views

plane cubic with a singularity must have non-constant morphism from $\mathbb{P}^1$?

If $C$ is a plane projective curve which is defined by an irreducible homogeneous cubic polynomial and has a singularity, why must there be a nonconstant morphism $\mathbb{P}^1\rightarrow C$? (I'm ...
4
votes
3answers
64 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 $\mathbb R^{n+1} \setminus \{0\}$ ...
4
votes
1answer
60 views

Expressing $\mathbb{R} P^3$ as a fibre bundle

This question came up in office hours with my differential topology prof and since then I've almost settled on an answer. The question was whether we could write $\mathbb{R} P^3$ as a fiber bundle ...
4
votes
1answer
138 views

Consequences of Properness in Algebraic Geometry

Let's call a variety $X$ proper if the projection $Y\times X\rightarrow X$ is a closed map (where $Y$ is any variety). I read in Vakil's notes that properness is a version compactness in algebraic ...
4
votes
1answer
193 views

The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
4
votes
1answer
120 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 ...
4
votes
3answers
106 views

The Lie algebra of the generators of the projective transformation is isomorphic to the Lie algebra of traceless matrices.

The general projective transformation of the $x$-$y$ plane is given by $$\tilde{x}=\frac{a_1x+a_2 y+a_3}{a_7x+a_8y+a_9},\quad\tilde{y}=\frac{a_4x+a_5y+a_6}{a_7x+a_8y+a_9}$$ for some constants ...
4
votes
1answer
97 views

Segre varieties contained in hyperplanes

Recall, the Segre embedding is a map $\sigma: \mathbb{P}^m \times \mathbb{P}^n \to \mathbb{P}^{(m+1)(n+1)-1}$ given by \begin{equation} ([x_0:\cdots:x_m],[y_0:\cdots:y_n]) \mapsto [x_0y_0:x_0y_1: ...
4
votes
1answer
2k 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 ...
4
votes
2answers
102 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 ...
4
votes
1answer
117 views

Identifying a line bundle on $\mathbb{P}^1$

I have a geometric line bundle $L$ on $\mathbb{P}^1 = \{[x_0:x_1]\}$. With respect to the standard affine cover $U_0 = \{x_0 \neq 0\}$ and $U_1 = \{x_1 \neq 0\}$, I have the transition function ...
4
votes
1answer
221 views

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 ...
4
votes
1answer
149 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 ...
4
votes
1answer
106 views

some fun with holomorphic line bundles

These are probably trivial questions... (for the experts) I'd like to get convinced (perhaps an intuitive/geometric explanation will be more effective than a formal one) of the following facts: i. ...
4
votes
1answer
135 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 ...
4
votes
0answers
48 views

How to find equations that define the image of an algebraic morphism?

Suppose we have a map $f:\mathbb{P}^n\rightarrow \mathbb{P}^m$ which is algebraic. What are the techniques to find the equations defining the image of f as a subvariety of $\mathbb{P}^m$? For example ...
4
votes
0answers
111 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 ...
4
votes
0answers
200 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
4
votes
0answers
107 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 ...