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3
votes
0answers
196 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
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 ...
7
votes
1answer
502 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 ...
18
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
363 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
242 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
227 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
577 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
423 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
580 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
374 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
229 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 ...
1
vote
1answer
470 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
623 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
660 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
222 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
219 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
809 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
498 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 $P$ ...
7
votes
2answers
504 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 ...
3
votes
2answers
983 views

Function Field of projective space

I'm reading the book of Silverman about elliptic curves. It describes the function field of a variety defined over K to be the quotient field of K[X]/I(V), then says that we may look at the function ...
5
votes
4answers
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 ...
6
votes
4answers
436 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. ...
3
votes
2answers
375 views

Flag Varieties - Projective Curves

Is it true that there are no projective curves which are also flag manifolds? If so, why?
15
votes
6answers
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 ...
29
votes
11answers
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 ...
9
votes
9answers
3k views

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?