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5
votes
1answer
490 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
768 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
362 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
197 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 ...
2
votes
1answer
531 views

Line bundles, linear systems and normalization

One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on ...
2
votes
2answers
454 views

Components of an exceptional divisor

Let $X$ be a projective variety and let $\tilde{X}$ be the blow-up of $X$ at a subscheme $Z$. Let $F$ be the exceptional divisor of $\tilde{X}$. I wonder: What is the number of irreducible ...
2
votes
2answers
1k views

Generalisations of Riemann-Roch for surfaces

Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have $$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$ This is the famous ...
2
votes
1answer
247 views

intersecting sections on the projective line

This question is about intersection theory on the easiest (arithmetic) surface over $\mathbf{Z}$: $\mathbf{P}^1_{\mathbf{Z}}$. Suppose we are given two distinct $\mathbf{Q}$-rational points $b_1$ and ...
2
votes
3answers
1k views

How to solve geometry problems using involutions

Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in ...
1
vote
1answer
513 views

projective geometry question

what is the three parameter family of plane projective transformations which fix a unit circle at the origin(that is map the unit circle to itself)? I understand that one such transformation is a ...
29
votes
10answers
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 $2$. ...
1
vote
1answer
198 views

Projecting projective curves

I've been stuck for quite a while on what is probably a trivial problem. Let $X\subset\mathbb{P}^n$ be a smooth projective curve, and let $$\mathcal{I}=\{(p,q,r):p,q\in X,p\neq ...
2
votes
2answers
2k views

Calculate camera position from 3x4 projection matrix

I have a 3 x 4 projection matrix $P$ given that calculates a homogeneous 2-Vector ${\bf i}=(u,v,w)^T$ on some screen (e.g.) from a homogeneous 3-Vector ${\bf x}=(x,y,z,w)^T$ in world space by $P \cdot ...
4
votes
2answers
307 views

Reference on the Veblen-Young characterization of projective spaces

Can someone point me to a modern treatment of the Veblen-Young characterization of projective spaces of dimension greater than $2$ as $P(V)$ for some vector space $V$? [Added: see here for a ...
21
votes
2answers
2k 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 ...
14
votes
2answers
984 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 ...
7
votes
4answers
474 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 ...
0
votes
2answers
190 views

Dimensionality of a map — distance

Hello, I am looking for some words to describe what going on here. I'm sure this is not an original thought, so I'd like to read up on more from others who have thought out this topic further. FORMAT ...
3
votes
0answers
198 views

Polarizations on $M_{0,n}$ from Kapranov's quotient constructions

In Kapranov's marvelous paper Chow quotients of Grassmannian I, he proves that $\overline{M}_{0,n}$ is isomorphic to both the Hilbert quotient and Chow quotient $(\mathbb{P}^1)^n//\text{SL}_2$. These ...
8
votes
1answer
379 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 ...
7
votes
1answer
504 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 ...
19
votes
1answer
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 ...
9
votes
3answers
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 ...
5
votes
2answers
366 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 ...
4
votes
0answers
248 views

Quaternionic Veronese Embedding

I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow \mathbb{C}P^3$ is ...
1
vote
0answers
230 views

The geometry of PSO(4) and the quaternions [closed]

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
6
votes
2answers
589 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 ...
2
votes
0answers
1k views

Automorphisms of projective space [closed]

According to Wikipedia, Aut(P(V)) = PGL(V). Apparently this is proved by using sheaves generated by global sections but I'm not familiar with this notion. I would appreciate it if anyone could ...
0
votes
1answer
426 views

geometric interpretation of componentwise linear fractional transformation(LFT)

Let x,y,z be points taken exclusively from the positive orthant. For the scaling transformation x'=x/(x+y+z) y'=y/(x+y+z) z'=z/(x+y+z) where each function is a linear fractional transform how can I ...
0
votes
1answer
285 views

Convex sets and projections

Hello! I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...
8
votes
2answers
587 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
votes
2answers
378 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 ...
27
votes
2answers
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 ...
61
votes
6answers
6k 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 $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual ...
2
votes
0answers
235 views

quasi-trigonal curves

I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...
2
votes
1answer
479 views

Blocking set in a projective plane.

Let S be any nontrivial blocking set in a projective plane of order q, such that S not containing any line. Let A be a set of points in the same projective plane of order q, raging over all these S. ...
4
votes
2answers
258 views

Endomorphisms of bundles associated to codimension 2 subvarieties

Preamble I initially decided to post this question on math.stackexchange a few days ago, as I consider it to be much less of a research question and much more of "I'm learning" question. But there ...
2
votes
2answers
627 views

Projective transformation between polygons.

Extending my earlier question about linear transformations, what's the easiest way to test if there exists a projective linear transformation that takes one polygon to another (in ...
3
votes
2answers
665 views

Decomposition of GL(2,p) into irreducible representations

Let $G=GL(2,p)$ be the group of linear transformations. It acts on the set $X=F_p^2$. Then $C^X$ is a linear representation of $G$. What is the decomposition of this representation into irreducible ...
1
vote
1answer
224 views

References on the dual frame of a projective frame.

A projective frame $\mathcal{F}$ of a projective space $P=P(E)$ defines a basis of the vector space $E$, defined up to multiplication by a scalar. The corresponding dual basis of $E^*$ is well defined ...
0
votes
1answer
220 views

Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)}

$S$ is a graded ring (over non-negative integers), $f \in S_{+}$ is a homogeneous element of positive degree, $D(f)$ the elements of Proj $S$ not containing $f$. I don't see the bijection between ...
2
votes
1answer
820 views

How far is the tangent bundle from projective space?

Is there a general theory of embeddings of the (total variety of) the tangent bundle on a (nonsingular) projective variety into projective space? I suppose what I really mean is (and to be more ...
2
votes
2answers
511 views

What is known about finite morphisms from X to the projective line

Let $f:X\longrightarrow \mathbf{P}^1$ be a finite morphism of schemes. If $f$ is etale and $X$ is connected, we can show that $f$ is an isomorphism. That is, the projective line is simply connected. I ...
23
votes
6answers
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 ...
3
votes
1answer
151 views

Edge-maximizing projective transformation on polytopes

Let P and Q be simple polytopes such that P = Q ∩ H and let H be a halfspace with normal vector n. Let projn(e) denote the length of the projection of edge e onto vector n. Consider the set E of ...
4
votes
1answer
186 views

Rectifying texture from image

I have a camera matrix $P$ which defines a projective transformation $\mathbb{P}^3 \rightarrow \mathbb{P}^2$. In the former space there is a plane $[ x|\pi^Tx=0 ]$. The image of the plane under $P$ ...
8
votes
2answers
533 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 ...