Questions tagged [projective-geometry]
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620
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Understanding homogeneous projective rational manifolds
Every complex flag manifold is a homogeneous projective rational manifold (HPRM). But are there examples of HPRMs that are not complex flags? Can somebody provide examples of such spaces? For some ...
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A paper of Borel (in German) on compact homogeneous Kähler manifolds
I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German
Satz I: Jede zusammenhängende kompakte homogene ...
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A standard name for the algebraic structure on a projective line?
Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...
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Are quadruples $abcd$ and $dcba$ always projectively equivalent in any projective plane?
It is well-known that for every line $L$ in a Pappian projective plane (i.e., a projective plane over a field) and any distinct points $a,b,c,d\in L$ the quadruples $(a,b,c,d)$ and $(d,c,b,a)$ are ...
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Borromean rings on $\Bbb{RP}^2$ and octonions
If I draw a trefoil knot on a projective plane and draw a circle around it touching the three outer parts of the curve. I can view this as a division of the projective plane in 8 triangles, viewed as ...
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Why do all incidence theorems follow from Pappus' theorem?
In Hilbert and Cohn-Vossen's ``Geometry and the Imagination,"
they state in the last paragraph of Chapter 20 that "Any
theorems concerned solely with incidence relations in the
[Euclidean projective]...
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Extending the proof of Maschke's Theorem from finite groups to algebras
In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
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Does the Affine Pappus Axiom imply the Affine Desargues Axiom in affine planes?
I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian).
First I introduce all necessary definitions.
Definition L. A ...
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A corollary of the affine Desargues axiom
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
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A graphic representation of classical unitals on 28 points
I would like to understand the geometry of the classical unitals.
They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I ...
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Computing $\pi_1$ of the complement of a non-singular plane curve
The following is a well-known fact:
Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.
This was further ...
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A formula for the cross-ratio in terms of hyperbolic data
Let $(\zeta_i) \subset \hat{\mathbb{C}}$, for $i = 1, \ldots, 4$, be $4$ distinct points on the Riemann sphere $\hat{\mathbb{C}}$.
We will use the following convention for the cross-ratio $CR$ of ...
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Necessary and/or sufficient condition for invertibility of the gradient of a polynomial of $m$ variables, viewed as a self map of $\mathbb{R}^m?$
I was wondering whether the following is true, and if not, is something known in this direction?
Let $P:\mathbb{R}^m \to \mathbb{R}$ be a degree $r$ polynomial (not necessarily homogeneous) that ...
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Projective camera: back-projecting a point on the image plane into 3-space [closed]
Suppose I got a projective camera model. For this model I would like to back-project a ray through a point in the image plane. I know that the equation for this is the following:
$$
y(\lambda) = P^+_0 ...
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Surfaces with rational double points
Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it.
Blowing-up the two points and ...
CommunityBot
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Integer solutions to $x^2 + x + 1 = y^z$? [duplicate]
In the context of finite projective planes I am interested in the Diophantine equation $\frac{x^3-1}{x-1} = y^z$, which is also written as $x^2 + x + 1 = y^z$, for $z>1$. I stumbled by accident on ...
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Is there an underlying explanation for the magical powers of the Schwarzian derivative?
Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function
$S(f) = \frac{f'''}{f'} - \frac{3}{2} \Big(\frac{f''}{f'}\Big)^2$
Here is a somewhat more ...
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Mapping a cube to a sphere
I have been looking for a way to map a unit cube (with vertices $x^2=1$, $y^2=1$, $z^2=1$) to a unit sphere ($x^2+y^2+z^2=1$) with minimal distortion of the great circles formed by mapping the ...
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Picard group of quasi-projective varieties
Let $X$ be a smooth open sub-variety of a projective, not necessarily smooth, variety $X'$, defined over a finite field.
Is $\text{Pic}(X)$ a finitely generated abelian group?
I'm tempted to just ...
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Fermat cubic hypersurfaces over finite fields
Consider the Fermat cubic
$$
X = \{x_0^3+\dots +x_n^3 = 0\}\subset\mathbb{P}^n_{\mathbb{F}_{q}}
$$
over a finite field $\mathbb{F}_{q}$ with $q$ elements.
If $q \equiv 2 \mod 3$ then the projection $\...
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Projective Plane of Order 12
I asked this question on the new Theoretical Computer Science "overflow" site, and commenters suggested I ask it here. That question is here, and it contains additional links, which I doubt I can ...
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A projective plane in the Euclidean plane
Problem. Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is ...
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Moufang identities and Moufang plane
Moufang identities
$$x(y⋅xz)=(xy⋅x)z,$$
$$(zx⋅y)x=z(x⋅yx),$$
$$xy⋅zx=x(yz⋅x)$$
are identities deeply related with alternativity (since setting $z=1$ one recovers left and right alternativity), while a ...
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Singularities of fibrations in conics
Consider a rank two vector bundle $E = \mathcal{O}(a)\oplus \mathcal{O}(b)\oplus \mathcal{O}(c)$ over $\mathbb{P}^1$. Fix coordinates $u_0,u_1$ on the base $\mathbb{P}^1$ and $v_0,v_1,v_2$ on the ...
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Cohomology of a stratified projective bundle
Let $S$ be a smooth algebraic variety, and suppose $X\to S$ is a smooth morphism of schemes such that the geometric fibers are all projective spaces. Let us suppose that the dimension of the fibers is ...
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Centralness of planar convex regions - behavior under projective transformations
Background:
For a planar convex region $C$ and an interior point $P$ we define:
the centralness ratio at $P$ is
$$\min\left(\frac{\text{shorter portion of }\chi}{\text{longer portion of }\chi}:\chi\...
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Does the real part of the cross ratio satisfy a maximum principle on a domain in any real submanifold?
Let $C(p_1, p_2; p_3, p_4)$ denote the cross-ratio of the $4$ points $p_i$, for $i = 1, \ldots, 4$, thought of as a holomorphic function on
$$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times ...
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Mori cones and projective morphisms
Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties, and $NE(X),NE(Y)$ the Mori cones of curves of $X$ and $Y$. Assume that $NE(X)$ is finitely generated. Then is $NE(Y)$ finitely ...
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Galois action on blow-ups related to field extensions of infinite degree
Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...
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Computing the Pushforward for Arbitrary Coherent Sheaves in the Context of the Segre Embedding and K-Theory
Following thinking about a question from math overflow (and answering it https://math.stackexchange.com/a/4686391/299848) I was wondering about the topic:
Given the Segre embedding $\sigma: \mathbb{P}^...
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What is $(C, D, \delta, \gamma)$ and $(C, \delta; D, \gamma)$ Desarguesian?
A projective plane is $(C, \gamma)$-Desarguesian if for any 2 triangles $A_1 B_1 C_1, A_2 B_2 C_2$ in perspective from $C$ (which means $C \in A_1 A_2, B_1 B_2, C_1 C_2$) such that $A_1 B_1 \cap A_2 ...
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Polytope of a projected toric variety
I was looking for such a result in the book by Cox, Little and Schenck but I'm not able to find a proper reference.
All of the following requirements are tacitly assumed to be in the projective ...
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Linear span of tangential variety
Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$. Let us denote with $TX=\bigcup_{x \in X}\mathbb{T}_xX$ the tangential variety, where $\mathbb{T}_x X$ is the projective tangent ...
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Linear system giving the projective embedding of the tangential variety
I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this ...
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How to express this hyperbolic extension of the cross-ratio in terms of hyperbolic distances and volumes?
Given $4$ distinct points on the Riemann sphere, thought of as the sphere at infinity of hyperbolic $3$-space $H^3$, one may define the cross-ratio in the usual way. Note that the cross-ratio is the ...
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Exterior product of Euler Exact Sequence
Consider the Euler exact sequence:
$ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $
This ...
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Definition of canonical pair
Let $(X,D)$ be a pair and $f:Y\rightarrow X$ a log resolution. Write
$$
K_Y + \widetilde{D} = f^{*}(K_X) + \sum_{i}a_iE_i
$$
where $\widetilde{D}$ is the strict transform of $D$. I found the following ...
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Normal bundle of a linear subspace
Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...
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Computing the automorphism scheme of projective space
$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...
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Classification of quartic surfaces
Let $k$ be a field of characteristic zero (non necessarily algebraically closed, we may assume for instance that $k = \mathbb{C}(t)$). Does there exist a classification of degree four surfaces $S\...
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Example showing that $\mathbb{P}^1$ does not preserve monics
Is there an injective homomorphism of commutative rings $A \to B$ such that the induced map $\mathbb{P}^1(A) \to \mathbb{P}^1(B)$ is not injective? Here, $\mathbb{P}^1(A) = \mathrm{Hom}(\mathrm{Spec}(...
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Geometry of contracted divisors
Let $f:\mathbb{P}^3\dashrightarrow\mathbb{P}^2$ be a dominant rational map defined over a field $k$ (not necessarily algebraically closed) of characteristic zero.
Consider a resolution $\widetilde{f}:...
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Hypersurface of singular plane cubics
In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...
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Automorphisms of projective spaces, and the Axiom of Choice
It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly ...
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Does this hexagon theorem have a name?
Question : Do you know this property of a hexagon?
Consider the configuration: Six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ in a plane and let six points $B_i \in A_iA_{i+1}$ for $i=1, 2,\dots, ...
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How do we define the type of a singularity on a cubic surface?
Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the ...
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Is there a non-singular cubic surface that has a point where four lines intersect?
Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt ...
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Which finite projective planes can have a symmetric incidence matrix?
As the title says. Which finite projective planes admit a symmetric incidence matrix?
I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F_q)$ can always ...
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Semi-continuity of the Picard number
Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...
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A spherical geometry claim related to the perspective 3-point problem
I have a simple claim in spherical geometry that has come out of my research into the so-called "perspective 3-point (pose) problem."
Here it is:
Fix three (distinct) great circles on the ...
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