The projective-geometry tag has no wiki summary.

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### Projective planes that contain the Fano plane

I am interested in the following question:
Given a projective plane of order $n=2^a$, is its incidence matrix must contain the incidence matrix of the Fano plane? If not, is it true that for any $n$ ...

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### 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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319 views

### On well-formedness of weighted projective spaces and a Hurwitz theorem calculation

This question has two parts: A calculation that is giving me a lot of troubles, and a theoretical one on weighted projective spaces.
1) I want to find the genus of the curve $C_7 \subset ...

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115 views

### On the Haar measure of Grassmanians

How can one write down the Haar measure of complex Grassmanians in terms of Plucker coordinates? Is there any way to define a Kahlarian measure like $d\mu\propto \det(g)dp_{ij}dp^{*}_{ij}$ where $g$ ...

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### Angles and projective metric

Unless I am very wrong, the following seems to be true:
If the angle between two vectors in $\mathbb{R}^{n}_{++}$ is small, then the
value of the Hilbert projective metric between them is also ...

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277 views

### What criteria are to determine if two projective varieties are projectively equivalent?

A projective transformation is a morphism of $P^n$ to $P^n$, for some $n$, determined by an
$(n + 1) \times (n + 1)$ invertible matrix $A$ in the obvious way. The sets $Q$, $R$ are projectively ...

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847 views

### Visualization of the real projective plane [closed]

Consider a closed (compact and without boundary) and non-orientable 2-manifold $M$. By Whitney embedding theorem, one can embed $M$ in $\mathbb{R}^4$. $M$ cannot be embeded in $\mathbb{R}^3$ and just ...

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105 views

### A certain property of elliptic curves in a paper by Rees

In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski.
Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ ...

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270 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 ...

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276 views

### 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

186 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

665 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

411 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

107 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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287 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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199 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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250 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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239 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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202 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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217 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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119 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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429 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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80 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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265 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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577 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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390 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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266 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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242 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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### 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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337 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

264 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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### 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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154 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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200 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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### 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 many ...

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170 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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### 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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### 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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295 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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310 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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233 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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### 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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242 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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### 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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### 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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### 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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### (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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### 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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291 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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### 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 ...