The projective-geometry tag has no wiki summary.

**60**

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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
$S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$
Here is a somewhat more conceptual ...

**29**

votes

**11**answers

3k views

### What is the Cayley projective plane?

One can build a projective plane from R^n, C^n and H^n and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as OP^2, the Cayley projective ...

**29**

votes

**10**answers

2k 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 $2$. ...

**25**

votes

**2**answers

2k 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 ...

**23**

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**6**answers

2k 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 ...

**21**

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**2**answers

1k 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 ...

**18**

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**6**answers

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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 ...

**18**

votes

**1**answer

1k 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 ...

**16**

votes

**1**answer

1k 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 ...

**16**

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**1**answer

621 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), ...

**15**

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**6**answers

2k 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 ...

**15**

votes

**1**answer

413 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 ...

**14**

votes

**2**answers

954 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 ...

**13**

votes

**1**answer

532 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 ...

**13**

votes

**1**answer

548 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 ...

**11**

votes

**3**answers

747 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 ...

**11**

votes

**3**answers

301 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 ...

**11**

votes

**1**answer

588 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 ...

**10**

votes

**1**answer

361 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 ...

**10**

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**0**answers

255 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 ...

**9**

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**3**answers

1k 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 ...

**9**

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**1**answer

396 views

### $6$ points lie on a conic if and only if $ABC$ and $A_0B_0C_0$ are perspective

Let $ABC$ be a triangle with incircle $\omega$. Let $A_0,B_0,C_0$ be points outside $\omega$. The tangents from $A_0$ to $\omega$ intersect $BC$ at $A_1,A_2$. Define $B_1,B_2$ and $C_1,C_2$ similarly. ...

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**9**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?

**8**

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**1**answer

377 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 ...

**8**

votes

**2**answers

348 views

### Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$

Let us consider the points
$$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\
p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$.
...

**8**

votes

**2**answers

373 views

### When can one extend a flat family from a subscheme to the whole scheme?

Is there a nice condition on a closed subscheme $Y$ of $X$ such that for every flat family $Z\to Y$, there is a flat family $W\to X$ whose restriction to $Y$ is $Z$? In particular, I'm interested in ...

**8**

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**1**answer

238 views

### Existence of different knots in $RP^3$ having the equivalent liftings in $S^3$

I'm looking for the answer to following question. Do exist different knots in $RP^3$ which have equivalent liftings in $S^3$ under covering $p:S^3\rightarrow RP^3$?

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**2**answers

574 views

### Base locus of divisors on blowings up of the projective space

Let $X$ be the blowing-up of $\mathbb{P}^n$ in $r$ points in general position.
Let $\{H,E_1,...,E_r\}$ be the standard basis of $Pic(X)$. Further let $D=dH-\sum m_j E_j$, be an effective divisor, with ...

**8**

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**0**answers

162 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 ...

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**2**answers

500 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 ...

**7**

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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 ...

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**2**answers

306 views

### Minor theorems of Pappus and Desargues in “old school” geometry?

My question concerns the dependence relations between the minor theorem of Pappus which, following Heyting, I will denote by $P_9$, and (one of the) minor theorems of Desargues, $D_9$.
$P_9$ states ...

**7**

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**4**answers

472 views

### Low rate c-uniform pairwise intersecting set systems

Let $U$ be some (unbounded) universe of elements, and let $\mathcal{S}$ be a collection of subsets of size $c$ each, such that any two elements from $\mathcal{S}$ have a non-empty intersection. Let $C ...

**7**

votes

**1**answer

500 views

### Normality of a locus of points in projective space

Let $U_{d,n}\subseteq(\mathbb{P}^d)^n$ denote the locus of $n$-distinct points in projective space $\mathbb{P}^d$ that lie on a rational normal curve of degree $d$, and let $V_{d,n}$ denote its ...

**6**

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**4**answers

434 views

### Divisors, extensions of functions

This might be a silly/obvious question. I know that we have the removable singularity theorem (of Riemann) on the complex line, and we also have the generalization of this to algebraic curves. ...

**6**

votes

**2**answers

570 views

### Embedding of algebraic surfaces

If I am not mistaken there is a theorem that says any curve $C$ can be embedded in $\mathbf{P}^3$. What can be said about surfaces? Do we have a theorem like:
All surfaces can be embedded in ...

**6**

votes

**3**answers

957 views

### Original proof of Pappus' Hexagon Theorem

Does anyone know where I can find an english translation, preferrably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of Pappus' hexagon ...

**6**

votes

**1**answer

522 views

### Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...

**6**

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403 views

### Reference for Weighted Projective Stacks

For a sequence of positive integers $a_0, \ldots, a_n$ and a ring $R$, there is a graded ring $R[x_0,\ldots, x_n]$ where $x_i$ is in degree $a_i$. There is a corresponding $\mathbb{G}_m$-action on ...

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**1**answer

534 views

### Proper morphism sending coherent to coherent

Hello,
Is there a proof that the push forward by a proper morphism of Noetherian schemes sends coherent sheaves to coherent ones, without passing in the argument through projective morphisms?
Thank ...

**6**

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**0**answers

110 views

### Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...

**5**

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**4**answers

784 views

### Is the theory of incidence geometry complete?

Consider the basic axioms of planar incidence geometry, which allow us to speak of in-betweeness, collinearity and concurrency. These axioms per se are not complete, since for example, Desargues ...

**5**

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**2**answers

361 views

### Fano 3-fold of degree 4

Let $X$ be the intersection of two quadrics in $P^5$. It is well known that the intermediate Jacobian $J(X)$ is isomorphic to $J(C)$ for a genus 2 curve, related to the pencil of quadrics whose base ...

**5**

votes

**1**answer

550 views

### Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...

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**2**answers

433 views

### Basics(?) about quasi-coherent modules on projective schemes

EDIT. (05-04-12) I have revised and improved the questions.
Let $A$ be a commutative $\mathbb{N}$-graded $R$-algebra, which is finitely generated by $A_1$ as an $A_0$-algebra. You may also assume ...

**5**

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**1**answer

462 views

### Are cubic four-folds containing a quartic scroll pfaffians?

Let $X\subset \mathbb{P}^5$ a smooth pfaffian smooth cubic fourfold hypersurface. It is easy to see that such a hypersurface must contain a quartic scroll surface. I wonder about the inverse question. ...

**5**

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**2**answers

345 views

### Generalization of Pascal's Theorem to Higher Dimensions

Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...

**5**

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**1**answer

95 views

### Measuring how closely a missing projective plane can be approached by an equivalent structure

It is well known that for a number of structures, their existence is equivalent to the existence of a projective plane for a given order. Some of them depend on more than one parameters, which means ...

**5**

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**0**answers

171 views

### Non minimal K3 surfaces as hypersurfaces of weighted projective spaces

I recently learnt that the hypersurface
$$
S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1)
$$
is birational to a K3 surface. This is surprising because the surface is quasi-smooth, ...

**4**

votes

**2**answers

499 views

### How do I find the set of all lines lying on a general quadric in $\mathbb{CP}^3$?

I have heard that this set is the disjoint union of two conics in $Gr(2,4)$, but I do not have an original reference. Does anyone either have such a reference, or know a way of seeing this?