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-2
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24 views

Virtual Christmas Tree Project [on hold]

Introduction Even though it is April, that doesn’t mean we can’t get in the spirit and begin to prepare for the big celebration next December. You may not be aware of this but, every year there is a ...
2
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0answers
18 views

Automorphisms of Complex Projective Space

Each automorphism $\mathbb{C}P^n\rightarrow \mathbb{C}P^n$ (where $\mathbb{C}P^n$ is regarded as a complex manifold) is induced by a linear map $\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$. I know ...
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0answers
31 views

What is wrong with my statement?

X,Y elements of Projective Space P. I got this wrong from the start and need clarification: X is dependent iff there is a finite subset X' subset of X which is dependent. I started with this: If X' ...
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0answers
20 views

Generalization of a projective plane?

In the area of finite geometry, a projective plane is an incidence structure of points and lines with the following properties: Every two points are incident with a unique line Every two lines are ...
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1answer
18 views

Why does it take 5 points to construct a projective frame in $\mathbb{R}^4$?

I am reading this answer http://math.stackexchange.com/a/186254/130408. The original question in that post is about deducing the projection matrix. But I have difficulties in understanding that ...
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21 views

An exercise in Gathmann's lecture notes about Projective spaces

EX. 3.5.1 in Gathmann's lecture notes: Let $L_1$ and $L_2$ be two disjoint lines in $\mathbb{P}^3$, and let $P\notin \mathbb{P}^3 \setminus (L_1 \cup L_2)$ be a point. Show that there is a ...
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0answers
31 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 ...
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1answer
50 views

Computing the Todd class of projective space.

As an exercise I'm trying to verify that for $X=\Bbb{P}_k^n$, where $k$ is an algebraically closed field, we have $$\operatorname{td}(X)=\left(\frac{\epsilon}{1-e^{-\epsilon}}\right)^{n+1},$$ where ...
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1answer
30 views

How can I prove formally that the projective space is a Hausdorff space?

I want to prove the Hausdorff property of the projective space with this definition: Define $\mathbb{P}^n$, the real projective space of dimension n to be the set of 1-dimensional linear subspaces ...
0
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2answers
85 views

orthogonal projection and Cauchy Schwarz inequality

Show that if P is an orthogonal projection matrix, then $||Px||\le||x||$ for every x. Use this inequality to prove the Cauchy Schwarz inequality. I know that if P is an orthogonal projection matrix, ...
3
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1answer
61 views

Imagining the projective Space

I am trying to get used to work in the projective space. Therefore I wanted to know which tactics there are to imagine the projective space. $$\mathbb{P^n}(k):= (k^{n+1}\backslash \{0\})/k^{*}$$ I ...
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0answers
50 views

Graded projective modules and vector bundles on projective varieties

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded ...
4
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1answer
81 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. ...
3
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1answer
50 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 ...
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0answers
32 views

Construction of projective space of a graded ring

Let $S=k[x_0,x_1,x_2]$, $k$ an algebraically closed field and $ S_d=k[x_0^d,x_0^{d-1}x_1,\ldots,x_2^{d-1}x_1, x_2^d]$. $\mathbb{P}S_d $ is identified with the projective space $\mathbb{P}^{N_d} $, ...
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1answer
48 views

Why is $(0,0,0)$ not acceptable as a co-ordinate on the projective plane?

Reading "Elliptic Tales: Curves, Counting, and Number Theory" which states $(0,0,0)$ cannot be a co-ordinate on the projective plane but I find the argument advanced for this in the book to be ...
2
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1answer
36 views

Intersection of two open sets in the projective plane

I want to compute the cohomology groups of the real projective plane, $P^2$, using Mayer Vietoris exact sequence. Now $H^0(P^2)=\mathbb{R}$, $H^2(P^2)=0$ being $P$ not orientable, so my problem really ...
2
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1answer
52 views

vector field on $\mathbb{R} P^2$

Actually this is a quesion in Lee's book, Manifolds and differential geometry. I have problems working with projective spaces as manifolds.(e.g. what are curves in projective spaces ? I need to know ...
3
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1answer
93 views

Isomorphism in homology of $\mathbb{R} P^2$

I have a question about the homology of the real projective space $\mathbb{R} P^2$ with which I'm having some trouble: Let $f: \mathbb{R}P^2 \rightarrow \mathbb{R}P^2$ be a map which induces an ...
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1answer
40 views

In the Proj construction, are the homogeneuos prime ideals determined up to an element in the base ring?

Consider the point $[a_0,...,a_n]\sim [a_0\lambda,...,a_n\lambda] \in \Bbb P^k_n$. How do you write the corresponding homogeneous prime ideal in the graded ring $S:=k[x_0,...,x_n]$? Well, the ...
0
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1answer
35 views

Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
1
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1answer
77 views

Canonical line bundle over a projective bundle

The following is an excerpt from the Atiyah's K-Theory. If $E$ is a vector bundle over $X$ then each point $a\in P(E)_{x}=P(E_{x})$ represents a one-dimensional subspace $H_{x}^{*}\subset E_{x}$. ...
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1answer
40 views

Sheaf cohomology of projective spaces

I came across Bott's formula in "Vector bundles on complex projective spaces" by Okonek, Schneider & Spindler. The formula is a formula for $h^q(\mathbb P^n,\Omega^p(k))$, where $\Omega^p(k)$ is ...
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0answers
26 views

Questions on linear subspace of a projective space

I am a bit confused by the definition of the linear subspace of a projective space. It says in a book "Algebraic Geometry: A first course" by Joe Harris on page 5 that An inclusion of subspace ...
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1answer
44 views

Constructing image under homography from known information

If $f$ is a homography of the real projective line, $f^2=id$ (is an involution), and $f$ has exactly two fixed points, how can I construct (geometrically) the image of an arbitrary point?
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19 views

Projective homologies

In a projective plane (i.e. two-dimensional) $\mathbb P$, we call a general homology a projective transformation $h:\mathbb P\to\mathbb P$ such that $h$ has a line of fixed points $L$ called the axis ...
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0answers
38 views

Classification of Projective transformation according to Jordan form

Say we have the projective space $\mathbb P^2_{\mathbb R}$ = $\mathbb P(\mathbb R^3) \stackrel {\text{def.}}{=} \{\text{span(u)}\mid u\in\mathbb R^3\smallsetminus\{0\}\}$. Denote $[u]$ for an element ...
3
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1answer
101 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 ...
6
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1answer
125 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 ...
2
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2answers
64 views

“Intrinsic” treatment of projective spaces

The question I would like to ask is the following one. Consider a projective space just as a smooth manifold, e.g. $\mathbb{C}P^1$ is $S^2$. Then most maps from $S^2$ to $S^2$ even if smooth, ...
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1answer
47 views

Affine stratification of Grassmannian $\mathbb{G}(1,\mathbb{P}^3)$

Let $G=\mathbb{G}(1,\mathbb{P})$ be the Grassmannian variety of lines in $\mathbb{P}^3$. I have to do an affine stratification of $G$. In order to do this we consider the flag $\mathcal{F}$ of the ...
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1answer
85 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 ...
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1answer
36 views

Are eigenspaces in a Hilbert space rays?

It may sound as a dumb question but I just want to be sure that I understand all the terminology: The eigenspaces corresponding to a (non-degenerate) eigenvalue of a operator on a Hilbert space are ...
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0answers
56 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
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1answer
110 views

proof regarding zeta function of a curve from Ireland and Rosen's “A Classical Introduction to Modern Number Theory”

In chapter $11$ section $5$ titled "the last entry" the authors state that the number of solutions to the congruence $x^2+y^2+x^2y^2 \equiv 1 \mod p$ is $p+1-2a$, where $p=a^2+b^2$ and $a+bi \equiv ...
2
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1answer
118 views

Hilbert polynomial of an hypersurface in projective space

Let $X$ an hypersurface in $\mathbb{P}^{n}$ of degree $d$. I would like to prove that the Hilbert polynomial of $X$ is $\qquad \qquad \qquad \qquad \qquad \qquad p(n)= \begin{pmatrix} n+r \\ r ...
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1answer
59 views

Homogenization of an affine variety in projective space

I think I have a fundamental understanding issue. I am trying to prove that if we embed $\mathbb{A}^{n}$ into $\mathbb{P}^{n}$ by $x_{0} \neq 0$, and we have affine algebraic sets $V(I) \subset V(J)$, ...
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0answers
52 views

What vector space is this?

Let $a,b,c$ be odd primes. In particular, $ab, ac, bc$ are all odd numbers. We can use this to our advantage, since then $\sqrt[ac]{x} : \Bbb{R} \to \Bbb{R}$ is well-defined and a bijection. Let ...
2
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1answer
82 views

On double covering of projective plane and map preserving antipodal points

Let $p:S^2\to P^2$ be the double covering of the real projective plane. Let $g:P^2 \to P^2$ be a map such that its induced homomorphism on fundamental group is not trivial. I'd like to show that ...
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0answers
18 views

How to extract projected points from Factor Analysis?

I am currently comparing different projection methods to derive 2D screen coordinates from high-dimensional point clouds. This is often done with PCA, MDS, LLE, SNE, etc. and I want to compare them ...
2
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1answer
57 views

Analytic subvariety in complex manifold

I am trying to figure out a statement in a textbook "If $M$ is any complex manifold of a projective space $\mathbb{P}^{n}$, $V\subseteq M$ an analytic subvariety of dimension $k$, then we can find a ...
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79 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 ...
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1answer
74 views

Blowup of $\mathbb{P}^n$ at a point is irreducible

The blowup of $\mathbb{P}^n$ at a point is irreducible. This seems clear intuitively, but I'm not sure how to prove it. Thoughts?
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1answer
21 views

the dimension problem of complex projection

Is it true that $\operatorname{dim }H^{0}(P^{n},P(T^{*}P^{n}))>0$? That is, is there a global holomorphic section? Here $P^{n}$ is $n$-dimensional complex projection space and ...
2
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2answers
182 views

Matrix projection on a cone

How would you project a symmetric real matrix onto the cone of all positive semi-de finite matrices?
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1answer
58 views

projection onto vector spaces

How do you project a vector on to the euclidean ball? For example, if there is a vector $x ∈ R^n$ how does one project this onto the euclidean ball. What are the steps for projecting a vector onto a ...
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0answers
13 views

Project a point to a box with an affine constrain in high dimension

Let a point be $p\in \mathbb{R}^N$. The constrain are $0\le p_i\le C, C>0$ and $ p^Tl=0$. I know if there is no box constrain. Then it would be $p^*=(1-\frac{ll^T}{l^Tl})p$. But if I have to ...
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0answers
124 views

Real-life example of Steiner's theorem

In projective geometry, Steiner's theorem reads: "Let $A,B$ be two different points of the projective plane and let $s$ be the line joining them. If $f:A^*\rightarrow B^*$ is a non-perspective ...
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17 views

Getting used to projective coordinates, need help describing (2) objects geometrically [duplicate]

I'm trying to get an intuition for what things look like in projective coordinates. There are two curves that I have to work a problem with, but I'm not sure how to visualize them. They are $V(u^2 X ...
2
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1answer
76 views

$V(X^m + Y^m - Z^m)$ (projective Fermat curve) isomorphic to projective line iff $m=1, 2$

I've convinced myself that the projective Fermat curve $V(X^m + Y^m - Z^m) \subset \mathbb{P}^2$ is isomorphic to a projective line if and only if $m =1$ or $m = 2$, but I'm not sure how to prove this ...