The projective-geometry tag has no usage guidance.

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### why is the homological projective dual of this Lefschetz decomposition non-commutative?

I am reading these notes of an excellent course by Kuznetsov on Homological Projective Duality. On page 10 there is Example 1''.
One starts with projective space, endowed with the identity embedding ...

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**1**answer

193 views

### Koszul complex of a variety inside a product

Suppose $X$ is a smooth projective complete intersection contained in the product $\mathbb{P}^n \times \mathbb{P}^m$, and call $X_n$ and $X_m$ the images of $X$ inside $\mathbb{P}^n$ and ...

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**1**answer

779 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), ...

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**1**answer

466 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 ...

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**1**answer

111 views

### How to visulize surface link in four dimension?

I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...

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**2**answers

309 views

### Positively curved manifold with a codimension 1 totally geodesic submanifold.

Fact : Consider the inclusion $V^{n-1} \rightarrow M^n$ where $M$ is a closed orientable simply
connected positively curved manifold.
Then connectivity lemma implies that the inclusion is ...

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**0**answers

212 views

### canonical model of a reducible curve

Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...

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**2**answers

261 views

### Was Desargues more an Euclid or an Eudoxos?

In the course of preparing lessons on projective geometry I want to give an account on the historical development. It is easy to obtain an overview of the history starting with G. Desargues. And with ...

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**1**answer

248 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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**1**answer

255 views

### A little question on certain parallel-lines-preserving maps

Let $\alpha:\mathbb{R}^n\to\mathbb{R}^n$, $n\geq 2$, be a $\mathbb{Q}$-linear bijection with the following properties:
1) $\alpha$ sends straight affine $\mathbb{R}$-lines to straight affine ...

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220 views

### Proper subgroups of $\rm{SU}(d)$ that act transitively on $\rm{CP}^{d-1}$?

The special unitary group $\rm{SU}(d)$ has a canonical action on the Hilbert space of dimension $d$, and this action induces a canonical action on the projective space $\rm{CP}^{d-1}$, which is ...

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122 views

### Flatness of projective bundles

Suppose that $p \colon P \to X$ is a projective bundle. Is $p$ an open map ?

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**3**answers

540 views

### On duality on finite projective planes

Hey Everyone!
In nearly all (if not all) projective geometry texts I have bumped into the following theorem:
"Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for ...

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**0**answers

84 views

### sphere with projective structure

In " Geometric structures on low-dimensional manifolds " , section 2 , we have :
given a projective tame 3-manifold with radial ends , each end surface has a projective structure since a developing ...

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**0**answers

288 views

### Dual of a weighted projective space

I have a fairly good understanding of what the dual of a projective space is. I am currently interested in weighted projective space but I haven't found anything on the construction of its dual space ...

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**2**answers

623 views

### A Problem about affine transformation

Problem: Suppose that $f:\;\mathbb{R}^2\to\mathbb{R}^2$ is an injective mapping from the 2-dimensional Euclidean plane into itself which maps lines into (instead of onto) lines and whose range ...

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**1**answer

407 views

### Independent generic/general points over some prime field

The first paragraph of this question shows the construction of the first counter example to Hilbert's 14th Problem. There, we start from a prime field $P$ of arbitrary characteristic, i.e., $P=\Bbb Q$ ...

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308 views

### What is the ideal corresponding to the Plücker embedding?

Let $S$ be a noetherian scheme, $\mathcal{E}$ a quasi-coherent sheaf on $S$ and let $d \in \mathbb{N}$. There is a Plücker embedding $\omega : \mathrm{Grass}_d(\mathcal{E}) \hookrightarrow ...

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**1**answer

284 views

### finite surjective morphism to the projective line

Let X a smooth projective curve over $\mathbb{C}$.
We fix $d$ distinct closed points $x_{1},\dots,x_{d}$.
Can we find a finite surjective morphism $\pi:X\rightarrow\mathbb{P}^{1}$
and local ...

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**1**answer

682 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 ...

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**1**answer

359 views

### (3,3) abelian surface and k3 surfaces

SOrry for the very specific question, but curiosity bites....
So here's the story: an idecomposable principally polarized abelian surface is embedded in $P^8=|3\Theta |^* $ as a deg 18 surface A. ...

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**1**answer

289 views

### octic K3s inside cubic 4-folds

From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could ...

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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 ...

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**0**answers

159 views

### Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...

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206 views

### complement of a codimension-one projective subspace

The complement of a codimension-one projective subspace of $\mathbb{R}\mathbf{P}^{3}$ is identifiable in a geodesic structure preserving manner with an affine $3$-space so that the group of projective ...

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683 views

### What is projective duality from modern point of view ? (correspondence ? Fourier on D-mod ? Aut of D(Coh) ?)

Consider vector space $V$ and its dual $V^*$ then to any line subspace in $V$ one can correspond its kernel in $V^*$ which is hyperplane.
Projective duality states that this correspondence satisfies ...

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174 views

### lefschetz theorem for quadrics

Does there exist an analogue of Lefschetz Hyperplane Theorem for cohomology that holds for intersections with (smooth) quadrics?

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**2**answers

564 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?

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824 views

### When does a planar ternary ring uniquely coordinitise a projective plane?

From every projective plane a coordinitisation can be constructed on a planar ternary ring, and conversely from every planar ternary ring a projective plane can be constructed. (For background see ...

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323 views

### Galois group decomposition of non-cyclic covers

If $\pi: C \rightarrow \mathbb{P}^{1}$ is a cyclic cover of $\mathbb{P}^{1}$ with Galois group $\mathbb{Z}/m \mathbb{Z}$ and thus with the (affine) formula
$y^{m}= ...

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331 views

### Equations for abelian coverings of $\mathbb{P^{1}}$

Cyclic coverings of $\mathbb{P^{1}}$ have a simple (affine) equation, namely the formula,
$y^{m}= (x_{1}-a_{1})^{t_{1}}....(x_{n}-a_{n})^{t_{n}}$. Is there such a nice equation for abelian non-cyclic ...

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254 views

### Projective spaces with nonconstant regular functions

I can construct a scheme by patching that represents a projective space over an arbitrary ring. I can also prove that, if the ring is a Jacobson domain, the only regular functions on it are constants.
...

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178 views

### What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...

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**2**answers

306 views

### Mapping multivariate polynomial inequalities system to subspace

What I will ask, more than a solution, is better mathematical definition of my problem and directions to find the solution.
I have a set of linear equations, e.g.:
\begin{align}
d_1 &= L_1 - ...

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1k views

### Geometric interpretation of the exact sequence for the cotangent bundle of the projective space

Edit: As Dan Petersen pointed out, this question is a duplicate of a previous one. I would leave it for the moderators to decide if this should be closed. On the other hand, may be this should be ...

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**0**answers

89 views

### invariant lines avoiding fixed subvarieties

Could anybody help me with the following question ?
Assume we are given:
(1) a finite order (linear) automorphism $g$ of the complex projective space $\mathbb{P}^r$,
(2) a closed algebraic ...

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459 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 ...

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**3**answers

829 views

### (Second) Chern class of projective space, blown up in a linear subvariety

I already asked the same question at stack exchange but got no response for quite a while, so I thought I'd ask here. I also know that this has a certain resemblance to this question, but I cannot ...

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**1**answer

236 views

### In what sense is a generically submersive morphism of varieties subermersive over singular points?

Background/Motivation
I'm currently interested in the duality theorem for projective varieties and more specifically in properties of the conormal variety over the dual variety.
Let $V$ be a ...

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**1**answer

292 views

### Are there n polynomials for which all intersection multiplicities are at least m?

I don't know whether this is known or not, but I was thinking of the following problem.
Let $n$ and $m$ be natural numbers. Are there $n$ polynomial $f_1,...,f_n\in \mathbb{C}[x]$, such that all of ...

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**6**answers

3k 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 ...

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**2**answers

611 views

### projective camera: back-projecting a point on the image plane into 3-space

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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1k 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 ...

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**1**answer

226 views

### How to construct S(5,8,24)

I am aware of one construction technique, involving 8-dimensional subspaces of a 24-dimensioinal vector space to create the octads. This technique is shown in 12 Sporadic Groups.
However. I am ...

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**1**answer

628 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 ...

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**1**answer

578 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 ...

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**1**answer

296 views

### A weird action of SL_3 on a pair of lines

Let us consider the complex projective plane $P^2$ and two distinct lines $L,L'\subset P^2$. Let us moreover consider the restriction of the natural action of $SL_3$ to $L\cup L'$. Can you tell in ...

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552 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. ...

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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 ...

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**2**answers

896 views

### Ample divisors on blown-up projective space

Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...