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

learn more… | top users | synonyms

0
votes
1answer
9 views

Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...
0
votes
0answers
15 views

find all straight-lines that goes through a point $p$ in projective and affine space [on hold]

I have an exam tomorrow in projective Geometry and I am lacking of many basic skills. For instance this one: How can I find all straight lines that go through $p$ in the projective space $\Bbb ...
0
votes
0answers
22 views

PGL_n transformation fixing a set of n+2 points in general position must be identity

I was wondering if anybody can think of a slick and short argument to show that any $PGL_n(k)$ transformation $A$ that fixes a set of $n+2$ points $p_1, p_2,\dots, p_{n+2}$ in general position must be ...
0
votes
1answer
35 views

Self-intersection number in Projective Space

The question is based on the example given in Intersection Theory under the heading Self-intersection. The example is as follows: Consider a line $L$ in the projective plane $\mathbb{CP}^{2}$: it has ...
1
vote
1answer
28 views

Darboux coordinates on projective spaces

I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of ...
0
votes
1answer
28 views

Irreducibility of an affine variety in an affince space vs in a projective space.

Proposition 5.5 in Undergraduate Algebraic Geometry by Reid says (I only write down a brief idea since the proposition is long and involves some other notations to define): The affine variety $U$ ...
0
votes
1answer
29 views

$U_{0}=\{[a^{0},a^{1},…,a^{n}]\in \mathbb{R}P^{n}\,:\,a^{0}\neq 0\}$. Is it open in $\mathbb{R}P^{n}$? [closed]

Is $U_{0}$ an open neighborhood of a $[a^{0},a^{1},…,a^{n}] \in \mathbb{R}P^{n}$ ? How can I prove that? It tried to see if $\pi^{-1}(U_{0})$ is open in $\mathbb{R}^{n+1}\smallsetminus\{0\}$. I see ...
0
votes
0answers
27 views

The degree of smooth projective curve included in $\mathbb P^n$ which is nondegenerate is more or equal the dimension of the projective space

I had a theorem during lecture, with proof which I don't understand. Theorem says: $ X \subset \mathbb{P}^n(\mathbb{C})$ smooth projective curve which is nondegenerate (not contained in a ...
1
vote
1answer
28 views

Number of points on a line in a finite projective plane

I've been reading some proofs regarding finite projective planes of order n, and often they start out by assuming that each line contains n+1 points. Is this a fact that follows from the axioms for ...
1
vote
0answers
24 views

Computing the inverse to a rational map

The setup: say I have some rational projective variety $X$ of dimension $n$ over $\mathbb{C}$ such that the map $$ X \dashrightarrow \mathbb{P}^n $$ is given by some linear series $\mathcal{L}$. My ...
5
votes
1answer
46 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 ...
0
votes
0answers
32 views

Computing the restriction $(\Lambda^2T_{\mathbb{P}^n})_{|L}$ for line in $\mathbb{P}^n$

From the Euler sequence $$0\to\mathcal{O}_{\mathbb{P}^n}\to V\otimes\mathcal{O}_{\mathbb{P}^n}(1)\to T_{\mathbb{P}^n}\to0$$ it is easy to deduce that ...
0
votes
1answer
50 views

Possible subbundles on $\mathbb{CP}^1$

I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why? We know that the ...
0
votes
0answers
17 views

Give $1^{st}$ order reduced model obtained by Galerkin projection, given the state space system

We have $\delta(t) = 1$ if $t = 0$ and $\delta(t) = 0$ if $t \neq 0$ and $x(t+1)= \begin{bmatrix} 1 &&{-0.5}\\0.5&&0 \end{bmatrix}x(t)+\begin{bmatrix} 1\\1\end{bmatrix}\delta(t)$ ...
1
vote
1answer
64 views

Computation of the fundamental group of $\mathbb{C}P^n$ using induction on $n$.

Let $n\geqslant 2$, I am asked to prove the following: Proposition. $\pi_1(\mathbb{C}P^n,\cdot)$ is isomorphic to $\pi_1(\mathbb{C}P^{n-1},\cdot)$. Proof. First, let us introduce some notation: ...
0
votes
2answers
57 views

Reference request on complex projective algebraic geometry

I am looking for a reference on complex algebraic projective geometry. Specifically, I would like to become more acquainted with notions like the dimension and the degree of a projective algebraic ...
0
votes
1answer
38 views

exercize about the foundamental group of $\mathbb{P}^n(\mathbb{R})$

Let $p$ be a point in $\mathbb{P}^n(\mathbb{R})$ and $\Sigma$ the set containing all the projective lines passing through $p$. Given $s\in \Sigma$ we can define a continous closed path (let's say ...
2
votes
1answer
46 views

Are the two standard descriptions of $\mathbb{C}P^{\infty}$ (topologically) equivalent?

While reading through some issues of Baez's (wonderful) "This Week's Finds in Mathematical Physics," I came across this statement (from week 149): $K(\mathbb{Z},2)$ is a bit more complicated: it's ...
0
votes
0answers
36 views

Computing the ramification index of a morphism of curves

Definition: Let $f: C_1 \to C_2$ be a nonconstant map of smooth curves and let $P \in C_1$. $$e_f (P) = \textrm{ord}_P (f^* t_{f(P)})$$ where $t_{f(P)} \in K(C_2)$ is a uniformizer at $f(P)$ ...
2
votes
1answer
83 views

Maps from Sum of Projective Planes to Circle

this is a problem from Lee's Topological Manifolds, 11-21. It asks the following: What compact, connected surfaces $M$ admit a non-nullhomotopic map to the circle? So, we use the classification of ...
1
vote
1answer
23 views

Can 3 transformations (V, Σ, U) of SVD to describe a perspective transformation?

As known SVD (Singular value decomposition) is a factorization of the form M = UΣV∗. https://en.wikipedia.org/wiki/Singular_value_decomposition SVD of the linear map T can be easily analysed as a ...
2
votes
0answers
15 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
3
votes
0answers
46 views

Global sections of Proj

In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to $\textrm{Spec } ...
3
votes
1answer
49 views

Punctured complex projective space

Let $\mathcal{P}\mathbb{C}^{n}$ be the complex projective space of $\mathbb{C}^{n+1}$, and let $B=\{\mathbf{e}_{1},\cdots,\mathbf{e}_{n+1}\}$ be a basis in $\mathbb{C}^{n+1}$. I would like to ...
3
votes
1answer
58 views

Which line bundle has transition function $\psi_{12}([z_1,z_2])=\frac{z_1/z_2}{|z_1/z_2|}$?

We know that the complex line bundles over $\Bbb{CP}^1$ are classified by the integers. Each is isomorphic to one of the bundles $\mathcal{O}(n)$ for $n\in\Bbb Z$, where $\mathcal{O}(-1)$ is the ...
0
votes
1answer
23 views

Metric in the projective space $P^n$

Let $S^n = \{x \in \mathbb{R}^{n+1}; \langle\, x,x\rangle = 1 \}$. $P^n$ is the set of all unordered pairs $[x] = \{x,-x\}$, $x \in S^n$. I'd like to prove that $d([x],[y]) = \min ...
3
votes
0answers
42 views

Bijective correspondence of rational points in projective space

Let $k$ be a field and consider an arbitrary point $\alpha = (\alpha_0, \dots, \alpha_n) \in \mathbf{P}(k^{n+1})$. Then there is a bijection $\rho: \mathbf{P}(k^{n+1}) \to \mathbf{P}_k^n (k)$, the ...
1
vote
0answers
27 views

Construction of relative projective space via glueing

I would like to gain further practice on glueing schemes by constructing projective space over a ring. I am considering the following: I wonder how we get that $\mathcal{O}_{X_i} (X_{ij}) = ...
1
vote
1answer
44 views

Effect on eigenvalues of a projection matrix when removing its main diagonal?

Do you know what happens to a projection matrix when you remove the main diagonal? Consider $P=Z(Z'Z)^{-1}Z'$ and form P-diag(P-{i,i}). Now, when you eigendecompose $P$ we get zeroes and ones as ...
1
vote
1answer
30 views

Dimension of the fiber of a projective variety

When looking at this I found the theorem Given a surjective map of quasi-projective varieties $\phi:X\rightarrow Y$ and $y\in Y$, we have that dim$(\phi^{-1}(y))=$dim$(X)$-dim$(Y)$. First I was ...
0
votes
1answer
31 views

Projective space, immersion, embeding

How can I check if the map $F:\Bbb{P}^2\to \Bbb{P}^5$, given as follows $F([x,y,z])=([x^2,y^2,z^2,xy,yz,zx])$, is smooth, an immersion or an embedding?
2
votes
0answers
25 views

Extension of projection from a point to a Blow Up

I feel like there's something obvious I'm missing here, and I'm not looking for a whole answer, but rather just a pointer in the right direction. Suppose you have the projection from a point ...
0
votes
0answers
44 views

Equivalence of a vector bundle being trivial on $\mathbb{P}^1$

I am looking for various statements about a vector bundle $E$ of arbitrary rank being trivial on the complex projective line, $\mathbb{P}^1$. In particular, some arguments about cohomology would be ...
1
vote
2answers
30 views

Projective Linear Group and Projective Space

my aim : Why $GL(V)/Z(GL(V))$ is termed as projective general linear group? The reason seen in books says This group acts on the projective set/space faithfully. But, there can be many groups acting ...
1
vote
0answers
28 views

Are $ \mathbb{A}^n (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles?

Are $ \mathbb{A} (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when $ k $ is a domain ? Thanks in advance for your help.
1
vote
1answer
28 views

Obtaining the Fundamental Polygon of $\mathbb{R}P^2$

On this page, Wikipedia shows, under the "Examples" heading, the fundamental polygons of the Sphere and the Real Projective Plane. Can we obtain the latter diagram from the former? I thought that this ...
1
vote
2answers
74 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
1
vote
0answers
21 views

twist and product of projective spaces

On $\mathbb P^n_k\times_k \mathbb P^m_k$, is it true that $T_{\mathbb P^n_k\times_k \mathbb P^m_k}\otimes \mathcal O_{\mathbb P^n_k\times_k \mathbb P^m_k}(d,e)\simeq p_1^*T_{\mathbb P^n_k}(d)\oplus ...
0
votes
0answers
12 views

Last Row of the perspective projection matrix

Could you explain to me what is the purpose of -1 in the last row of the projection matrix? And how it affects the perspective division step ?
1
vote
1answer
43 views

embeddings of projective spaces into Euclidean spaces

Let $\mathbb{R}P^n$, $\mathbb{C}P^n$, $\mathbb{H}P^n$ be the real, complex, quaternionic projective spaces resp. I want to find all $n$ such that $\mathbb{R}P^n$ can be embedded into ...
3
votes
1answer
66 views

Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of ...
1
vote
0answers
30 views

Number of independent coefficients of homogeneous polynomials

Consider $R_d=k[X_0,\ldots, X_n]_d$, the homogeneous polynomials of degree $d$ in $n+1$ variables. The dimension of $R$ viewed as $k$-vector space is $\binom{n+d}{d}$ so a general polynomial in $R$ ...
1
vote
0answers
29 views

Projective Geometry True/False Questions

These questions are with respect to the projective plane: An ordinary point cannot lie on the ideal line; An ideal point cannot lie on an ordinary line; Every ordinary line cuts the ideal ...
2
votes
1answer
92 views

Change of basis between coordinate charts in $\mathbb RP^2$.

I'm looking at the real projective plane $\mathbb RP^2$, with homogeneous coordinates $(x:y:z)$. I want to find the transition matrix between bases associated with different coordinate charts. I've ...
2
votes
0answers
66 views

Global sections of projective subspaces

I have a general question with a specific application. The cohomologies of line bundles on $\mathbb{P}^n$ are known and in particular, $H^0(\mathbb{P}^n, \mathcal{O}(d))$ is canonically isomorphic to ...
0
votes
0answers
24 views

Can a point have a nontrivial isometry group?

This question is extremely related to this other question. In fact, a positive answer here directly implies a positive answer there. However, since it is a mathematically different question I decided ...
0
votes
1answer
52 views

punctured real projective space

Let $\mathbb{R}P^m$ be the real projective space and $X=\mathbb{R}P^m\setminus \{*\}$ be the punctured space by removing one point. How to get the cohomology ring of $X$ with integer coefficient? Is ...
0
votes
1answer
31 views

Calculating Hom in derived category

I got stuck calculating $Hom^* (\mathcal O, \mathcal O(k)) \in D(Coh(\mathbb P^n))$. On one hand, $Ext^i (\mathcal O, \mathcal O(k)) = H^i (\mathcal O^* \otimes \mathcal O(k)) = H^i (\mathcal O(k))$, ...
0
votes
0answers
23 views

Topology on $\Bbb{P}$ such that every affinity h : $\Bbb{P}\setminus{H}\to\Bbb{R}^n$ is a homeomorphism

I have the following excercise: "Let $\Bbb{P}$ be a real projective space of dimension n. Define on $\Bbb{P}$ a topology such that for every projective hyperplane H, every affine transformation h : ...
0
votes
1answer
24 views

Are inhomogeneous coordinates … COORDINATES?

It might seem a silly question but I'm asking the following: Take the complex projective line, are the inhomogeneous coordinates sufficient to have an atlas where the transition functions are ...