Questions tagged [projective-geometry]

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136 votes
9 answers
19k views

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 ...
Paul Siegel's user avatar
  • 28.8k
46 votes
11 answers
6k views

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 $\Bbb OP^2$, ...
skupers's user avatar
  • 7,923
42 votes
2 answers
2k views

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 $A\otimes_k B$ ...
George Bergman's user avatar
41 votes
2 answers
5k views

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 ...
Aaron Sterling's user avatar
37 votes
1 answer
2k views

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 ...
Gro-Tsen's user avatar
  • 29.9k
32 votes
10 answers
3k views

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 $2$. ...
Jesus Martinez Garcia's user avatar
28 votes
6 answers
4k views

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 ...
Tom Smith's user avatar
  • 1,180
27 votes
2 answers
3k views

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 ...
Peter Scholze's user avatar
22 votes
11 answers
12k views

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?
Ilya Nikokoshev's user avatar
22 votes
5 answers
9k views

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 ...
Greg Muller's user avatar
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22 votes
6 answers
2k views

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 ...
Pierre's user avatar
  • 2,145
22 votes
6 answers
2k views

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 $...
Matt Pusey's user avatar
22 votes
1 answer
2k views

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), ...
Drew Armstrong's user avatar
21 votes
1 answer
2k views

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 ...
IMeasy's user avatar
  • 3,717
20 votes
1 answer
2k views

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?...
Dan Petersen's user avatar
  • 39.2k
19 votes
2 answers
2k views

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]...
aaron's user avatar
  • 378
19 votes
1 answer
2k views

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 ...
Peter Scholze's user avatar
18 votes
1 answer
1k views

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 ...
Huangjun Zhu's user avatar
18 votes
3 answers
1k views

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 $IF$,...
Đào Thanh Oai's user avatar
17 votes
6 answers
7k views

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 votes
3 answers
1k views

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 ...
Seva's user avatar
  • 22.8k
16 votes
2 answers
785 views

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. ...
Jérémy Blanc's user avatar
16 votes
1 answer
695 views

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. ...
Joy Morris's user avatar
15 votes
2 answers
731 views

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 ...
მამუკა ჯიბლაძე's user avatar
15 votes
3 answers
2k views

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 ...
John Stillwell's user avatar
15 votes
4 answers
900 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 ...
Mike Shulman's user avatar
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) = ...
user76284's user avatar
  • 1,793
14 votes
1 answer
2k views

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 $k^n$....
Andrew's user avatar
  • 141
14 votes
1 answer
707 views

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 ...
Jeremy Shipley's user avatar
14 votes
1 answer
495 views

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?
user avatar
13 votes
2 answers
2k views

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 ...
Đào Thanh Oai's user avatar
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 ...
Asvin's user avatar
  • 7,648
13 votes
1 answer
780 views

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 ...
user avatar
12 votes
1 answer
307 views

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 ...
asv's user avatar
  • 21.1k
12 votes
1 answer
443 views

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 ...
coudy's user avatar
  • 18.5k
12 votes
1 answer
381 views

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 ...
byu's user avatar
  • 666
12 votes
2 answers
782 views

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 ...
Gil Kalai's user avatar
  • 24.2k
12 votes
3 answers
400 views

(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 ...
Stanley Yao Xiao's user avatar
12 votes
1 answer
934 views

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 ...
E. Vargas's user avatar
  • 188
12 votes
1 answer
1k views

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 ...
Harry van Langen's user avatar
12 votes
0 answers
557 views

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(...
user81500's user avatar
  • 121
11 votes
3 answers
1k views

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 ...
Carlos Esparza's user avatar
11 votes
2 answers
1k views

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 ...
CuriousUser's user avatar
  • 1,420
11 votes
2 answers
521 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 ...
Puzzled's user avatar
  • 8,842
11 votes
1 answer
412 views

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 ...
Elsa's user avatar
  • 113
11 votes
1 answer
1k views

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 ...
Asvin's user avatar
  • 7,648
10 votes
3 answers
2k views

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 ...
Marc Palm's user avatar
  • 11.1k
10 votes
3 answers
869 views

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.
user avatar
10 votes
1 answer
443 views

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 ...
solbap's user avatar
  • 3,938
10 votes
1 answer
489 views

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 ...
Taras Banakh's user avatar
  • 40.8k

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