Questions tagged [projective-geometry]
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614
questions
136
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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 ...
46
votes
11
answers
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What is the Cayley projective plane?
One can build a projective plane from $\Bbb R^n$, $\Bbb C^n$ and $\Bbb H^n$ and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as $\Bbb OP^2$, ...
42
votes
2
answers
2k
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Is this lemma in elementary linear algebra new?
Is anyone familiar with the following, or anything close to it?
Lemma. Suppose $A$, $B$ are nonzero finite-dimensional vector spaces
over an infinite field $k$, and $V$ a subspace of $A\otimes_k B$
...
41
votes
2
answers
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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 ...
37
votes
1
answer
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When do 27 lines lie on a cubic surface?
Consider $27$ (pairwise distinct!) lines in $\mathbb{P}^3$ whose intersection graph is that expected¹ of the $27$ lines on a smooth cubic surface. Question: Is there a simple necessary and sufficient ...
32
votes
10
answers
3k
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Which 'well-known' algebraic geometric results do not hold in characteristic 2?
A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
28
votes
6
answers
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How should I visualise RP^n?
So I did some algebraic topology at university, including homotopy theory and basic simplicial homology, as well as some differential geometry; and now I'm coming back to the subject for fun via ...
27
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2
answers
3k
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Accumulation of algebraic subvarieties: Near one subvariety there are many others (?)
Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the ...
22
votes
11
answers
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What is the exact statement of "there are 27 lines on a cubic"?
I think there was a theorem, like
every cubic hypersurface in $\mathbb P^3$ has 27 lines on it.
What is the exact statement and details?
22
votes
5
answers
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Maps to projective space determined by a line bundle
The following should be pretty standard for any algebraic geometer.
Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for ...
22
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6
answers
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About the definition of E8, and Rosenfeld's "Geometry of Lie groups"
I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...
22
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6
answers
2k
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Does every ellipse inside a tetrahedron inside a ball fit in a triangle inside the ball?
In three-dimensional euclidean space, consider the closed unit ball $B$. Let $T$ be a tetrahedron, and $E$ an ellipse, with $E \subset T \subset B$. Does there necessarily exist a triangle $T'$ with $...
22
votes
1
answer
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Octonions and the Fano plane.
Does the Fano plane mnemonic for octonion multiplication have any deeper meaning?
http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg
The symmetry group of the Fano plane is PSL(2,7), ...
21
votes
1
answer
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Rationality of intersection of quadrics
Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking ...
20
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1
answer
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Maps to projective space == line bundles; what do maps to weighted projective space correspond to?
A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a weighted projective space?...
19
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2
answers
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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]...
19
votes
1
answer
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Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 3
Part 3 of this series of questions. In the meantime, I realized that there is some very simple question that was left open in Accumulation of algebraic subvarieties: Near one subvariety there are many ...
18
votes
1
answer
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When is $(q^k-1)/(q-1)$ a perfect square?
Let $q$ be a prime power and $k>1$ a positive integer. For what values of $k$ and $q$ is the number $(q^k-1)/(q-1)$ a perfect square, that is the square of another integer? Is the number of such ...
18
votes
3
answers
1k
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An ellipse through 12 points related to Golden ratio
I am looking for a proof of the problem as follows:
Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$,...
17
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6
answers
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Explaining the concept of projective space: notes for students
This is a question on teaching.
I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
16
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3
answers
1k
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Fano plane drawings: embedding PG(2,2) into the real plane
By a drawing of the Fano plane I mean a system of seven simple curves and
seven points in the real plane such that
every point lies on exactly three curves, and every curve contains
exactly three ...
16
votes
2
answers
785
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What are Sylvester-Gallai configurations in the complex projective plane?
A Sylvester-Gallai configuration in the the complex projective plane is a finite number of $n\ge 2$ points in the complex projective plane such that there is no line through exactly two of them. ...
16
votes
1
answer
688
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Transitive actions of finite subgroups of ${\rm GL}(n,\Bbb Z)$ on projective geometries
For any $n$, the group ${\rm GL}(n,\Bbb Z)$ has a natural action on $\Bbb Z^n$. Modding out a prime $p$ yields an action on the vector space $F_p^n$, where $F_p$ is the finite field with $p$ elements. ...
15
votes
2
answers
729
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Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space
I've got ten (projective) planes in projective 3-space:
\begin{align}
&x=0\\
&z=0\\
&t=0\\
&x+y=0\\
&x-y=0\\
&z+t=0\\
&x-y-z=0\\
&x+y+z=0\\
&x-y+t=0\\
&x+y-t=0
...
15
votes
3
answers
2k
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Natural examples of Reverse Mathematics outside classical analysis?
Harvey Friedman at the 1974 ICM motivated Reverse Mathematics by the
following statement:
When the theorem is proved from the right axioms, the axioms can be proved
from the theorem.
Reverse ...
15
votes
4
answers
897
views
Synthetic projective lines
The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and ...
14
votes
1
answer
549
views
Projective-invariant differential operator
This question was originally asked on Math StackExchange.
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align*}
&T(g) = ...
14
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1
answer
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Who first proved the fundamental theorem of projective geometry?
The following theorem is often called the fundamental theorem of projective geometry:
Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of $k^n$....
14
votes
1
answer
705
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Vocabulary of 19th Century analytic projective geometry: What are "order" and "dimension"?
I am trying to understand the following introductory passage in an early lecture by the philosopher/mathematician Gottlob Frege because I am interested in how Frege conceived of the role of geometric ...
14
votes
1
answer
491
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Birational automorphisms of varieties of Picard number one
Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?
13
votes
2
answers
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Is it a new discovery on conic section?
I discovered a problem in plane geometry (there are some nice special cases) as follows:
Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...
13
votes
2
answers
558
views
A geometric definition of the addition law on abelian surfaces
Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines.
Is there a ...
13
votes
1
answer
780
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Generalization of the rigidity lemma in birational geometry
Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...
12
votes
1
answer
305
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Geodesic preserving diffeomorphisms of constant curvature spaces
Let $X$ be either Euclidean space $\mathbb{R}^n$, the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$.
I would like to have a classification of all diffeomorphisms $X\to X$ which map ...
12
votes
1
answer
443
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How many subspaces are generated by three or more subspaces in a Hilbert space?
In the book of Garrett Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using ...
12
votes
1
answer
380
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Does every space curve lie on a rational surface?
Let $C\subset \mathbb P^3$ be a smooth projective curve over $\mathbb C$. Is there a (singular) rational surface $S$ such that $C\subset S\subset \mathbb P^3$?
I'm also interested in the following ...
12
votes
2
answers
781
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A question about pairs of lines in 3D projective space
Consider a 3-dimensional projective space $X$.
Let $m$ be the smallest number so that there are $m$ pairs of lines
$ \ell_1,\ell'_1$, $ \ell_2,\ell_2'$, ... , $\ell_m, \ell'_m$ in $X$:
a) For ...
12
votes
3
answers
400
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(Non-)Existence of curves of low degree on affine and projective varieties
I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...
12
votes
1
answer
934
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Is Wikipedia correct about desarguesian projective planes being self-dual?
I stumbled over a statement on Wikipedia http://en.wikipedia.org/wiki/Duality_%28projective_geometry%29 and would like to ask how this could possibly be true. It states the following
The projective ...
12
votes
1
answer
1k
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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 ...
12
votes
0
answers
556
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Embedding of the product of two Grassmannians into a Grassmannian
Consider an embedding $$\Phi: G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})\rightarrow G_k(R^n)$$ of the product of two Grassmannians $G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})$ into $G_k(R^n)$, where $G_k(...
11
votes
3
answers
1k
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Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?
Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X_0 ⊆ X_1 ⊆ X_2 ⊆ \cdots ⊆ X_k = \mathbb{P}^n$ whose dimension ...
11
votes
2
answers
1k
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Can projective hypersurfaces contain linear spaces? How big?
I am in this, rather friendly, situation:
I have a complex projective space $\mathbb{P}^n$, and there i have a (possibly non-smooth) hypersurface $S$ defined by one irreducible polynomial $P$ of ...
11
votes
2
answers
519
views
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 ...
11
votes
1
answer
412
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Planes in Lagrangian Grassmannians
Let $LG(h,2h)$ be the Lagrangian Grassmannian of subspaces of dimension
$h$ of a complex vector space of dimension $2h$.
For instace, $LG(1,2)=\mathbb{P}^1$, and $LG(2,4)\subset\mathbb{P}^4$ is
a ...
11
votes
1
answer
1k
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What is the automorphism group of the projective line minus $n$ points?
$\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed ...
10
votes
3
answers
2k
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Why can projective varieties just have abelian group operations?
I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is ...
10
votes
3
answers
869
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Automorphisms of cartesian products of curves
Let $C$ be a smooth projective curve. Is it true that
$$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$
and in case, what would be a reference for this? Thanks.
10
votes
1
answer
443
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Can curves differentiate vector bundles on P^2?
Not much is known about vector bundles on $\mathbb{P}^2$ but I wonder if the following is a tractable question:
If $E,E'$ are non-isomorphic vector bundles on $\mathbb{P}^2$, then is there always a ...
10
votes
1
answer
488
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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 ...