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2
votes
1answer
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 ...
3
votes
2answers
282 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 ...
0
votes
0answers
194 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) ...
4
votes
2answers
248 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 ...
8
votes
1answer
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 $p:S^3\rightarrow RP^3$?
1
vote
1answer
190 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 ...
3
votes
2answers
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 ...
0
votes
2answers
118 views

Flatness of projective bundles

Suppose that $p \colon P \to X$ is a projective bundle. Is $p$ an open map ?
4
votes
3answers
414 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 ...
0
votes
0answers
79 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 ...
4
votes
0answers
260 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 ...
4
votes
2answers
567 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 ...
4
votes
1answer
379 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., $P=\Bbb Q$ ...
3
votes
2answers
263 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 ...
1
vote
1answer
219 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 ...
11
votes
1answer
611 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 ...
4
votes
1answer
330 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. ...
2
votes
1answer
259 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 ...
18
votes
6answers
1k 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 ...
1
vote
0answers
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 ...
0
votes
1answer
199 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 ...
2
votes
0answers
463 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 many ...
0
votes
0answers
170 views

lefschetz theorem for quadrics

Does there exist an analogue of Lefschetz Hyperplane Theorem for cohomology that holds for intersections with (smooth) quadrics?
4
votes
2answers
507 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?
3
votes
5answers
753 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 ...
2
votes
0answers
293 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}= ...
2
votes
3answers
302 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 ...
0
votes
0answers
228 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. ...
1
vote
0answers
158 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 ...
1
vote
1answer
235 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 - ...
3
votes
2answers
856 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 ...
1
vote
0answers
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 ...
5
votes
2answers
439 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 ...
4
votes
3answers
723 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 ...
1
vote
1answer
197 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 ...
3
votes
1answer
290 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 ...
7
votes
6answers
2k 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 ...
-1
votes
2answers
519 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 ...
6
votes
3answers
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 ...
1
vote
1answer
221 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 ...
6
votes
1answer
542 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 ...
13
votes
1answer
557 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 ...
1
vote
1answer
291 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 ...
5
votes
1answer
487 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. ...
16
votes
1answer
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 ...
3
votes
2answers
757 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 ...
0
votes
1answer
192 views

Action of $SL_{n+1}$ on couples of linear spaces in $\mathbb{P}^n$.

Let $SL_{n+1}$ act on $\mathbb{P}^n$ in the natural way. Suppose I take two linear subspaces $\mathbb{P}^m$ and $\mathbb{P}^{n-m}$, with $m < n$, that intersect in one point. Is the action of ...
1
vote
2answers
361 views

Configuration space of flags

Let $U\subset \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be the Zariksi open set of ordered quadruple of distinct points in the projective line. The quotient of $U$ by the ...
3
votes
1answer
287 views

Sections of a fibration in intersections of quadrics

Suppose that we have a smooth variety $X$ of dimension $n$ that fibers (a flat morphism) over a curve $Y$, and s.t. the fibers of $X \to Y$ are all complete interesections of two quadric hypersurfaces ...
0
votes
0answers
193 views

Does the normalization of a projective morphism determine the line bundle?

Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms $$f:X \to \mathbb{P}^n$$ and $$g:X \to \mathbb{P}^m,$$ such that the image of $f$ is the ...