The projective-geometry tag has no wiki summary.

**3**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**2**answers

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

**6**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**6**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**4**answers

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

**4**answers

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

**2**answers

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

**6**answers

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

**11**answers

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

**9**answers

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?