Questions tagged [projective-space]
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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Intuition on fundamental group of $\mathbb R P^2.$
Show that $\pi_1 (\mathbb RP^2) \neq 0$ by demonstrating a loop in $\mathbb R P^2$ which does not lift to a loop in $S^2.$
Consider the two sheeted covering $p : S^2 \longrightarrow \mathbb R P^2,$ ...
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Tangent spaces of $\Bbb P^1$
Consider the following descriptions of the complex projective
line $\Bbb P^1$:
The unit sphere $\{(u,v,w) \in \mathbb{R}^3 \mid u^2+v^2+w^2=1\}$ which is identified with the Riemann sphere $\Bbb C \...
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Cup product structure on $\mathbb{R}P^2$
I don't understand a passage in my lecture notes. Here is the passage with my questions added in italics:
$H^\ast(\mathbb{R}P^2;\mathbb{Z}/2\mathbb{Z})$ is $\mathbb{Z}/2\mathbb{Z}$ in degrees $1$ and ...
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Prove that there is no holomorphic map $s: \mathbb{P}^n \to \mathbb{C}^{n+1} \setminus \{0\}$ with $p \circ s = \operatorname{id}$.
Let $p:\mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{P}^n$ be the natural projection. Prove that there is no holomorphic map $s: \mathbb{P}^n \to \mathbb{C}^{n+1} \setminus \{0\}$ with $p \circ s = \...
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Quotient of projective group scheme by a finite group action
Let $X$ be a projective group scheme, over some base $S$, and let $G$ be a finite group acting on $X$ by $S$-isomorphisms.
I would like to understand if/when the quotient $X/G$ is representable by a ...
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If $X$ is a simply-connected $n$-dimensional CW complex, then any $f : X \rightarrow \mathbb RP^{n+1}$ is homotopic to a constant map
Let $X$ be a simply-connected $n$-dimensional CW complex. The goal is to show that any (continuous) $f : X \rightarrow \mathbb RP^{n+1}$ is homotopic to a constant map.
Let $p : S^{n+1} \rightarrow \...
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If $f: \mathbb CP^2 \rightarrow \mathbb CP^2$ is a homeomorphism, then $f(\mathbb CP^1)$ intersects $\mathbb CP^1$ [closed]
Consider the standard embedding $\mathbb CP^1 \subseteq \mathbb CP^2$.
Let $f : \mathbb CP^2 \rightarrow \mathbb CP^2$ be a homeomorphism. The goal is to show that $f(\mathbb CP^1)$ always intersects $...
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Is it possible $S^n\cong \mathbb{RP}^n$? [closed]
I found that $S^1\cong \mathbb{RP}^1$. I wonder if this can happen for any dimension other than 1. I have inquired about it but have had no response....
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Tangents to a projective plane curve.
I'm studying William Fulton's "Algebraic Curves," and I'm currently studying projective plane curves. One of the problems in the section asks us to find the tangents to $xy^4+yz^4+xz^4$ at ...
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Geometric interpretation of dual Variety.
While I was reading the book, Discriminants, Resultants, and Multidimensional Determinants by Gelfand, Kapranov & Zelevinsky, I came upon the following statement:
Let $X \subseteq \mathbb{P}^n$ be ...
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Segre Embedding and lines on $x^2+y^2+z^2=1$ over $\mathbb{C}$
Let $x^2+y^2+z^2=1$ be the unit sphere over $\mathbb{C}$. Prove that the sphere has 2 rulings by straight lines.
I have learnt the Segre embedding $\mathbb{P}^1(\mathbb{C})\times \mathbb{P}^1(\mathbb{...
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Is there an algorithm for finding every isolated singular point on an algebraic variety, or a programming language that implements this?
Suppose one wishes to test if a given algebraic surface f(x,y,z,w) = 0 in projective 3 space has singular points, that is df/dx = df/dy = df/dz = 0, and one also wishes to calculate these singular ...
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Example of (maximal) projective spectrum
I read in a book the following definition of $Proj$.
Given $A=\bigoplus\limits_{n\geq0}A_n$ a $\mathbb{N}$-graded $\mathbb{C}$-algebra without nilpotent, we can define $$Proj(A)= \text{homogeneous ...
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Projectable contact vector fields of projective cotangent bundle
Consider the projective cotangent bundle $PT^\star M$ with the contact structure inherited from the kernel of the Liouville form of $T^\star M$. A contact vector field $X$ (infinitesimal ...
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Factor of a dual curve
Suppose that for a homogeneous polynomial $f(x,y,z)$ we have $f(x,y,z)=g(x,y,z) \cdot h(x,y,z)$. Assume $f_d$ and $g_d$ are the dual curves of $f$ and $g$ respectively. Is it true that $g_d$ is a ...
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Showing that the projective space is an n-manifold.
I have to show that the real projective space $\mathbb{RP}^n$ is an n-manifold. I will use the set label $\mathbb{P}^n$ instead of $\mathbb{RP}^n$. $\mathbb{P}^n$ the set of all the lines passing ...
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Understanding the CW decomposition of the real projective $n$-space
I'm trying to work on proving the following statement:
Let $\Bbb P^n$ be the $n$-dimensional (real) projective space. Then $\Bbb P^n$ has a CW decomposition with one cell in each dimension $0,...,n$, ...
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Proof Concerning homeomorphisms of $\mathbb{P}^2$
Is the following proof valid?
CLAIM:
The space obtained by attaching a disc to a Mobius Strip along the boundary is homeomorphic to the projective plane.
PROOF:
We begin by showing that the boundary ...
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Is It using basis change? Which is the way?
Let be $F : P^4(R)\rightarrow P^4(R)$ an homography with the matrix A. (P is the projective space)
$A = [[1,1,0,0,0], [0,1,1,0,0], [0,0,1,0,0], [0,0,1,2,1],[1,0,0,0,1]]$
I have the plane $Z : x_2 = ...
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Identifying sections when computing cohomology groups $H^i(\mathbb P^2,\mathcal O(d))$ via Cech cohomology
I want to compute the dimension and find generators of the cohomology groups $H^i(\mathbb P^2,\mathcal O(d))$. To do so I want to use Cech cohomology and compute directly, or at least try to. So we ...
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Mumford Regularity Theorem - Lazarsfeld's proof
I am studying Lazarsfeld's proof of Mumford's Regularity Theorem. At some point, under the assumption of $\mathcal{F}$ being $m$-regular, he considers the exact sequence
\begin{equation}
\cdots \...
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Why can't I apply subspace topology from $\mathbb{R}^3$ on real projective plane
In Crossley's Essential topology, the line about $\mathbb{R}P^2$ - It is not a subset of $\mathbb{R}^3$, since the elements of $\mathbb{R}P^2$ are not points in $\mathbb{R}^3$
but subsets of $\mathbb{...
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The complex projective space is isotropic [duplicate]
I have several times seen stated the fact that the complex projective space with the Fubini-Study metric is isotropic, but i can't seem to find the proof anywhere, can anyone suggest a book or answer ...
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Global sections of a vector bundle on the projective space
This comes from a problem I'm working on, but I would like to phrase it in generality.
I consider a vector bundle $E$ on the complex projective space $\mathbb{P}^2$. Is there some criterion to decide ...
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Does $\left(\mathscr{L},\left[s_0, s_1, \ldots\right]\right)$ correspond to morphism $X \rightarrow \mathbb{P}^{\infty}$?
Suppose $X$ is a scheme and $\mathscr{L}$ an invertible sheaf on it. Given a sequence $\left[s_0, s_1, s_2, \ldots\right]$ of global sections on it that doesn’t have common zeros, does it induce a ...
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base-point-free linear series induce morphism to projective space coordinate-freely (for infinite-dimensional case)? [duplicate]
Suppose $X$ is a $k$-scheme and $\mathscr{L}$ an invertible sheaf on $X$. Show that a (finite-dimensional) base-point-free linear series $V$ on $X$ correspondoing to $\mathscr{L}$ induces a morphism ...
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Can $S^1$ act freely on $\mathbb{C}P^2$?
Prove that $S^1$ cannot act freely on the complex projective space $\mathbb{C}P^2$.
I just learned some basic concept about homology group and covering space. But I have no clue how to prove this ...
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A result used in the proof of Bezout's theorem.
I am reading Fulton's book and I have reached the proof of Bezout's theorem which states that:
Theorem: If $F$ and $G$ are two projective plane curves of degrees $m$ and $n$ respectively and they ...
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incident variety as the projectivization (relative proj of symmetric sheaf) of some locally free sheaf on the dual projective space
For simplicity let's suppose $n=2$ and $\Bbb{P}^{n}_k = \operatorname{Proj}k[X,Y,Z]$ and $\Bbb{P}^{n\vee}_k = \operatorname{Proj}k[A,B,C]$. With the segre embeding
$$
\operatorname{Proj}k[X,Y,Z] \...
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Zero subscheme and zero locus
On this page on StacksProject about coherent sheaves on projective spaces they define a hypersurface to be the zero scheme $Z(s)$ of a global section $ s \in \Gamma(\mathbb{P}_k^n,\mathcal{O}(d))$, ...
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Is a projected subspace equivalent to a truncated subspace?
I am from physics with humble mathematical background so apologies if you find this trivial. Given a vector $v=\begin{pmatrix} a \\ \alpha \\ b \\ \beta \end{pmatrix}$that lives in $\mathbb{R}^{4\...
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Finding a unique projective subspace
Yesterday I started studying projective geometry and today I was making some exercices on this. The following problem is were I got stuck, it goes as follows: In $P(\mathbb{R}^{n+1})$ Let $P^r$, $P^q$ ...
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Relationship (if any) between jet bundles and projectivized vector bundles
I am learning jet bundle theory for my research in physics. I have read 6 (of 7) chapters of "The Geometry of Jet Bundles" (D. J. Saunders), but I am confused about the relationships between ...
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Embedding $\mathbb{CP}^n$ into $\mathbb{R}^{4n-1}$ [closed]
I'm looking for a generalization of the map of $\mathbb{C}P^1$ into $\mathbb{R}^3$ as a sphere.
Are there any such (preferably canonical) embeddings of $\mathbb{C}P^n$ into $\mathbb{R}^{4n-1}$? (And ...
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Understanding projective change of coordinates.
An affine change of coordinates on $\mathbb A^n$ ,the affine $n$-space is defined to be a polynomial map $T:\mathbb A^n\to \mathbb A^n$ which is given by linear homogeneous polynomials and is also ...
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Representation of point at infinity of elliptic curves in Affine space
I am self reading theory of elliptic curves over finite fields. The following question has come to my mind.
Let $E$ be Weierstrass form of an elliptic curve over $\mathbb{F}_p$, the point at infinity ...
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Irreducible components, dimension and degree of projective varieties
I have this problem given to me in my review session for my algebraic geometry final:
Describe the irreducible components and compute the degree and dimension of $V_p(x_0x_2-x_1^2, x_0x_3-x_1x_2)\...
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Unique projective line through any two distinct points in projective space
I’ve just started studying projective geometry and this is the first result we are proving. By the way, $V$ is a finite-dimensional vector space, and I write $\mathbb{P}(V)$ for the projective space ...
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Proving $\mathbb{R}P^n$ is orientable if and only if $n$ is odd.
Prove that $\mathbb{R}P^n$ is orientable if and only if $n$ is odd.
I know this question has been asked many times on this site, but all solutions consist of $n$ forms or homology groups which I can'...
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canonical divisor from canonical sheaf
Let $X$ be a complete intersection of $m$ hypersurfaces in $P^n$ over some field $k$. I have computed that the canonical sheaf is $\omega_X=\mathcal{O}_X(\sum d_i-n-1)$, where $d_i$ are the degrees of ...
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Homology of Lens spaces with coefficients in an abelian group, and explicit generators in some cases
Consider the $2n+1$ dimensional sphere $S^{2n+1}$ as the submanifold of $\mathbb{C}^{n+1}$ defined by
$$
|z_1|^2+...+|z_{n+1}|^2=1 \ .
$$
Given an integer $p$ we consider the $\mathbb{Z}_p$ action on $...
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Prove that $\pi_{2n+1}(\mathbb CP^n) \cong \mathbb Z$ using the mapping cone
Let $f : S^{2n+1} \rightarrow \mathbb C P^n$ be a continuous map and $X := \mathrm{Cone}(S^{2n+1}) \cup_f \mathbb C P^n$.
We have the following long exact sequence:
$$\cdots \rightarrow \pi_{2n+2}(X) \...
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Projection of quasi projective variety is proper (or finite, or closed)
I'm trying to prove that variety $$X=\{(A,C,B)\vert AC=CA=BC=CB=0, \text{rk}(C)\leq1, \det(A)=\det(B)=0\}$$ is irreducible. Here $C$ considered up to multiplication by constant i.e. projectivisation ...
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Quotient set with respect to another set
The usual definition I have seen for a quotient set of some set $S$ is with respect to some equivalence relation $\sim$, i.e. $S / \sim$, which means all elements $x$ and $y$ where $x \sim y$ are ...
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Proving that Fubini metric is well-defined
Let
$$\pi:S^{2n+1}\to S^{2n+1}/S^1=\mathbb{P}^n\mathbb{C}$$
be the Hopf fibration. I already know that $d_z\pi$ is a vector space isomorphism for every $z\in S^{2n+1}$ (when restricted to the ...
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Projective Space Decomposition?
I am reading 'A Royal Road to Algebraic Geometry' by Audun Holme and in 1.1 he mentions that by diving $\mathbb{P}^k_n$, the projective $n$-space space over a field $k$, over and over, we can ...
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Why is the set $\displaystyle \{ [z_1,z_2,z_3] \in \mathbb C P^2 : z_2^2 z_3 = 4 z_1^3 - g_2 z_1 z_3^2 - g_3 z_3^3 \}$ connected?
I am wondering why the set $\displaystyle \{ [z_1,z_2,z_3] \in \mathbb C P^2 : z_2^2 z_3 = 4 z_1^3 - g_2 z_1 z_3^2 - g_3 z_3^3 \}$ is connected, where $g_2$ and $g_3$ are some fixed constants (for ...
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Number of orbits of projective action
let $L/K$ be a finite field extension (if you want, assume $K=\mathbf Q_p$).
Do we know the number of orbits of the action of ${\rm GL}_2(K)$ on the projective line $\mathbf P^1 (L)$ ?
If $L/K$ is ...
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Pretending to be a line in a finite projective plane
Projective planes
I'm working with the following definition of projective plane here: we have a set $P$ of points, and a collection $L$ of subsets of $P$ called "lines" with the following ...
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Pushforward of the Segre embedding in K-theory
Fix $n$, $m\ge 1$, and let $d=\binom{m+n}{m}$ and $N=mn+m+n$. Consider the Segre embedding $\sigma:\mathbb{P}^m\times \mathbb{P}^n \hookrightarrow \mathbb{P}^{N}$, which has degree $d$. I'm trying to ...
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