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9
votes
0answers
22 views

Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
2
votes
1answer
42 views

Axioms vs Models in projective and hyperbolic geometry

I'm studying Projective Geometry. The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. ...
0
votes
0answers
33 views

Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows ...
3
votes
0answers
47 views

Short proof of Borsuk-Ulam's

By examining the singular cohomology ring with $\mathbb{Z}/2\mathbb{Z}$ coefficients, it is easy to see that if $n>m$ that there can be no map $f:\mathbb{R}P^{n}\to \mathbb{R}P^m$ that induces ...
1
vote
1answer
17 views

Unitary transformation of Fubini-Study metric

I am trying to solve a problem in Introduction to Complex Geometry by D. Huybrechts, question 3.1.6 which is the following: let $A\in GL(n+1, \mathbb{C})$ be a $\mathbb{C}$-linear transformation ...
1
vote
0answers
19 views

Cohomology of intersection of hyperplanes

let $X = H_1 \cap ... \cap H_d$ be a compact submanifold of $\mathbb{P}_N$ where the $H_i$ are hyperplanes. I want to compute $H^q(X, \mathcal{O}_{\mathbb{P}_N}(m)|X)$. I am pretty unexperienced in ...
1
vote
0answers
48 views

Exceptional coherent sheaves on $\mathbb{P}^n$ are vector bundles

Let $E$ be a coherent sheaf over $\mathbb{P}^n_k$. Coherent sheaf is called exceptional if $\operatorname{Hom}(E,E) \cong k$, and $\operatorname{Ext}^{> 0}(E,E) \cong 0$. How one can show that such ...
1
vote
1answer
32 views

Help on formalisation proof of the triviality of a kernel in Mayer-Vietoris

Consider the Mayer-Vietoris sequence for $\mathbb{RP}^2$, where the two open sets are $U:= \{ [x;y;z] \in \mathbb{RP}^2 | z \neq 0 \}$ and $V = \mathbb{RP}^2 \setminus [0;0;1]$. I've proved that $U ...
8
votes
8answers
119 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 ...
0
votes
0answers
13 views

Embedding into the projective space - tangent level injection

I am reading Griffiths and Harris, the section of embedding a manifold $M$ into projective space. Let $\mathcal{L}$ be a line bundle over $M$, with dim $H^0(M,\mathcal{L})=N$, and let $s_0,...,s_N$ be ...
4
votes
2answers
84 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 ...
3
votes
2answers
84 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 ...
0
votes
1answer
34 views

$\mathbb{R}P^2$ and its fundamental group by identification of edges of unity square

Suppose we identify edges of the the unity square $[0, 1] \times [0, 1]$, as in the picture: http://de.wikipedia.org/wiki/Datei:ProjectivePlaneAsSquare.svg Now to compute the fundamental group, for ...
1
vote
1answer
23 views

The projection operator defined by $(P_n(h) - h, v)_H = 0$

Let $V \subset H$ be separable Hilbert spaces with continuous embedding and suppose $\{v_n\}$ be a (non-orthogonal) basis for $V$. If we let $V_n = \text{span}(v_1, ..., v_n)$ and given $h \in H$ we ...
1
vote
2answers
40 views

Parallel Lines Intersecting in the Projective Plane

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that ...
0
votes
1answer
35 views

how to compute the distance between a matrix and its lower rank approximation?

I have a matrix $X$ and $Z$ a lower rank approximation of $X$ obtained using only few of the columns of $X$. I would like to have a measure of how distant are $X$ and $Z$. In particular I would like ...
0
votes
0answers
23 views

Normalize matrix so that its projection equals identity - which to use?

I wish to normalize a given matrix M, n by k, so that its projection matrix equals the identity. This is to speed up computations. The transformation/decomposition would look like this: Generate M by ...
1
vote
0answers
18 views

Saari's homographic conjecture and the actual definition of homography

By the wikipedia definitions found here and here, and especially by the definition implicit in this MSE post, it seems two images are homographic if they are renderings of the same set of points in ...
1
vote
0answers
17 views

Measures on $\mathbb{P}^1(\mathbb{C})$ and $\mathbb{P}^1(\mathbb{C}_p)$?

Is there a natural measure on the above spaces? Ideally, I would like a measure that is invariant under automorphisms of $\mathbb{P}^1$.
4
votes
1answer
141 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 ...
1
vote
1answer
63 views

Some questions about complex curves in $\mathbb CP^2$

I would like to ask for some clarifications in the following questions about complex curves. My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a ...
1
vote
0answers
30 views

Morphisms of quasi-projective varieties

Let $Y\subseteq \mathbb{P}^n(k)$ be a quasi-projective variety. By Görtz, Wedhorn (page 32, Proposition 1.65) in order to show that $$h:Y\to \mathbb{P}^m(k), y\mapsto (f_0(y):\dots :f_m(y))$$ is a ...
2
votes
1answer
78 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
2
votes
2answers
73 views

Projective varieties and irreducibility

The "modern"(schematic) definition of a projective variety is the following: Let $k$ be an algebraically closed field. A projective variety over $k$ is a closed subscheme of $\mathbb ...
1
vote
1answer
34 views

What is the line going through points $(5, 5, 5), (2, 2, 2) \in \mathbb{R^3}$ when mapped it is mapped to a point in the real projective plane?

So the real projective plane is homeomorphic under a function $f$ to $\mathbb{R^3} - (0, 0 ,0)$. Hence lines in $\mathbb{R^3} - (0, 0 ,0)$ become points in the the real projective plane. So what is ...
6
votes
1answer
173 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 ...
0
votes
1answer
26 views

Isomorphism between projective space and grassmanian $\mathrm{Grass}(1,n+1)$

I have to show that the projective space $\mathbb{P}^n$ is isomorphic to the grassmanian $\mathrm{Grass}(1,n+1)=\{V\subseteq\mathbb{R}^{n+1}:V\,\text{linear subspace,}\,\dim\,V\,=1\}$ as well as ...
2
votes
1answer
18 views

Vector space and its Projecctivized Space

Why is the co-dimension one subspaces are the points of $\mathbb P(V^{\vee})$. $V^{\vee}$ is the dual space of V and and $\mathbb P(V)$ is the projectivized space of V. $\mathbb P(V)= ...
3
votes
2answers
47 views

How to show that $f_* (\sigma)=\sigma$ where $f$ is mapping between projective spaces $\mathbb{R}\text{P}^3$

Suppose that $f:\mathbb{R}\text{P}^3 \to \mathbb{R}\text{P}^3$ is continuous mapping without fix points and let $\sigma$ be (some) generator of group $H_3(\mathbb{R}\text{P}^3)$. Prove that ...
0
votes
1answer
53 views

The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle

Let $z,z_1,z_2,z_3$ be four points on the extended plane. Their cross-ratio $(z,z_2,z_3,z_4)$ by definition is the image $Tz$ of $z$ under the Möbius transformation $T$ that sends $z_1,z_2,z_3$ to ...
1
vote
2answers
50 views

Enumerative projective geometry

I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, ...
3
votes
1answer
148 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
1
vote
1answer
19 views

The degree of a map between complex projective lines

Let $P$ and $Q$ be complex polynomials such that $\deg P=p$, $\deg Q=q$ and $\gcd(P,Q)=1$. How can I: show that $F(z)=\frac{P(z)}{Q(z)}$ defines a smooth map $\mathbb{C}P^1\to\mathbb{C}P^1$? ...
2
votes
1answer
202 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
0
votes
0answers
31 views

Dimensions of the cohomology groups of certain complicated space

Let contruct the space $X$. We take the complex projective space $\mathbb{C}P^2$, pick two points $p_1, p_2 \in \mathbb{C}P^2$ and remove two small, disjoint, open $4$-balls $B_j$ centered at $p_j$. ...
1
vote
1answer
37 views

How to define a “distance” from point to line in 3D projective space which is projectively invariant?

Since the concept of distance in Euclidean space is not invariant in projective space, that is , distance is invariant under Euclidean transformations but not under projective transformations, is it ...
0
votes
1answer
40 views

Matrix for orthogonal projection

Given was $v_1 = \begin{pmatrix} i\\0\\1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 0\\i\\1 \end{pmatrix}$. 1) I needed a orthonormalized basis $B$ of the sub-space built by $v_1, v_2$ and I got: $B ...
0
votes
0answers
21 views

Projective line bundles and blowing up

I'm trying to understand some facts about Chern classes. Looking up for some special examples I found that seems the total space of the projective line bundle $P(M\oplus M^{-1})\to \mathbb{CP}$ must ...
1
vote
1answer
45 views

Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$ where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
1
vote
1answer
22 views

Is this subset of $PSL(n,\mathbb{R})$ Zariski-closed?

For some non-identity element $[A]\in PSL(n,\mathbb{R})$ ($[A]$ being the class of $A\in SL(n,\mathbb{R})$) and linearly independent vectors $x,y\in\mathbb{R}^n$, let $[x],[y]$ denote the classes of ...
2
votes
1answer
99 views

What exactly does a Mobius Transformation do?

From what I understand, a Mobius transformation is of the form f(z) = $\frac{Az+D}{Cz+B}$ where A,B,C, and D may be real or complex What is f(z) doing to z exactly? And what are some of the ...
0
votes
1answer
34 views

How does a linear fractional function behave like a $2\times 2$ matrix?

So I did the math for this and got \begin{align*} A &= a_1a_2 + b_1c_2\\ B &= a_1b_2 + b_1d_2\\ C &= c_1a_2 + d_1c_2\\ D &= c_1b_2 + d_1d_2. \end{align*} My book does not talk ...
0
votes
1answer
174 views

What does it mean to call horizontal lines through O the “points at infinity” in real projective plane $RP^2$?

This is a picture from my book. I extended the line M to get a better idea of where $p_n$ is. It says the following: It is natural to call the horizontal lines through O the "points at infinity". ...
1
vote
0answers
41 views

Isometries and geodesics in projective plane using covering

We define a relation in the sphere by identifying the antipodal points, the quotient space obtained is the projective plane $\mathbb{P}^2$. Also, the quotient map ...
0
votes
0answers
32 views

A peculiar fact about 3-dimensional complex projective space

I'm working on a result for my master's thesis, that right now involves translating a proof I don't quite follow, to something that is a bit more in line with what I already know. We define ...
1
vote
0answers
43 views

Problem about $\mathbb{P}^3(K)$

Show that four skew lines in $\mathbb{P}^3$ have two transversals in common. I know that exist a quadric which contains three of the four lines....but i'm stuck EDIT: If the skew lines are ...
1
vote
1answer
27 views

Is there any definition for homogeneous rotations?

Most of the geometric transformations can only be represented into square matrices via homogeneous coordinates, e.g., translation and 3D rotations with axes not through coordindate system origin. ...
0
votes
0answers
32 views

A question about complex projective $n-$space

Let $\mathbb{P}^{n}(\mathbb{C})$ be the complex $n-$projective space and let $$ U_i=\{[x]=[x_0:\dots:x_n] \in \mathbb{P}^{n}(\mathbb{C}): x_i \ne 0\} $$ be a subset of $\mathbb{P}^{n}(\mathbb{C})$. I ...
2
votes
2answers
64 views

Finding singularities of a projective curve

For $w \in \mathbb{C}$ we define the projective curve $$p(x,y,z):= x^3+y^3+z^3+wxyz.$$ Now I have to find all $w \in \mathbb{C}$ for which the projective curve $p(x,y,z)$ is singular and show that ...
3
votes
0answers
28 views

Is there an embedding of projective varieties $\mathrm{Grass}(r,n)\hookrightarrow(\mathbb{P}^{n-1})^{\times r}$?

Let $k$ be an algebraically closed field, and let $r\le n$ be positive integers. Let $\mathrm{Grass}(r,n)$ be the projective variety of all $r$-dimensional planes in $k^n$. Notice that ...