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### Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...

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180 views

### Counting curves of degree 4 in $\mathbb{P}^{3}$

Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...

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86 views

### Embed the normalization of a curve in a larger space

I'd like to believe that this problem has a positive answer, but I don't know a nice reference. Actually I've never worked with embedded curves, so I apologize in advance if the question is too silly.
...

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190 views

### Curve of 3-secant lines

Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety ...

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118 views

### semi-orthogonal decompositions and embeddings

This is most likely a stupid question, but I am curious. It is well known that $\mathbb{P}^n$ has a semi-orthogonal decomposition
$$D^b(X)=\langle \mathcal{O},\dots,\mathcal{O}(n)\rangle$$
Suppose ...

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**2**answers

255 views

### Möbius transformation by 3 points in the Minkowski model

Goal
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
I know that a projective ...

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**1**answer

95 views

### Non-equivariant vector bundles over complex projective $N$-space

From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant.
I wonder, do there exist ...

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303 views

### (Non-)Existence of curves of low degree on affine and projective varieties

I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...

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**1**answer

101 views

### minimality/universality of the Springer resolution of a determinantal variety

Let $X\subset P^n$ be a singular determinantal variety and $S\to X$ its Springer resolution. Let $X'\to X$ another resolution of singularities (say, a blow-up). Does $S$ have some ...

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**1**answer

95 views

### Flag primitivity of the correlation group of classical projective planes.

We know that the full automorphism group of the $\pi_q = PG(2,q)$ acts imprimitively on the flags (all flags through a fixed point form a block). But, things change when we consider the action of full ...

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177 views

### What is the formula for the commutative multiplication on CP(infinity)?

There is a classic formula for maps $\mathrm{CP}(r) \times \mathrm{CP}(s) \to \mathrm{CP}(r+s)$ or maybe $r+s+1$ using Plücker coordinates - IF memory serves. In the limit we get the abelian ...

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503 views

### Degree of a smooth projective variety

Let $i_1:X \hookrightarrow \mathbb{P}^n$ and $i_2:Y \hookrightarrow \mathbb{P}^N$ be two projective schemes.
Let $f:X \to Y$ be a surjective projective morphism between smooth projective varieties ...

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173 views

### Non minimal K3 surfaces as hypersurfaces of weighted projective spaces

I recently learnt that the hypersurface
$$
S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1)
$$
is birational to a K3 surface. This is surprising because the surface is quasi-smooth, ...

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**1**answer

157 views

### Dimension of the global sections of the Serre twisting sheaf on a curve

Let, $C$ be a projective curve (not necessarily reduced), $i:C \to \mathbb{P}^n$ be a closed immersion. Does there exist a bound on/geometric interpretation of the dimension of ...

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**1**answer

238 views

### singularities of the dual variety of a surface

I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...

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**1**answer

198 views

### Extension of linear system

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on ...

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**1**answer

279 views

### Does every ample divisor “span” a hyperplane?

Let $X\subset\mathbb{P}^n$ be a smooth projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the ...

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316 views

### Minor theorems of Pappus and Desargues in “old school” geometry?

My question concerns the dependence relations between the minor theorem of Pappus which, following Heyting, I will denote by $P_9$, and (one of the) minor theorems of Desargues, $D_9$.
$P_9$ states ...

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257 views

### A question on infinitely many closed points on a smooth projective variety and their behavior under embeddings

While working on a research problem (algebraic cycles), I bumped into a question that I want to prove, though I couldn't yet prove. After several days of attempts, I realized that if the following ...

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338 views

### Partitioning the Projective Plane

Throughout this post, by projective plane I mean the set of all lines through the origin in $\mathbb{R}^3$.
Side Note: If there are more standard definitions for any of the ideas presented here, ...

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**1**answer

283 views

### blow-up along singular variety

Can somebody give me a nice example of blow-up of a smooth algebraic variety along a singular subvariety? Something I can do some exercise on and check the differences with a smooth blow-up. Thanks!

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318 views

### Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...

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191 views

### Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces

For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric ...

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234 views

### Global sections of a coherent sheaf in terms of a presentation

Let $A$ be a graded ring satisfying the usual finiteness conditions of EGA II (for example $A_0$ is noetherian, $A_1$ is finite over $A_0$ and $A$ is generated by finitely many elements of $A_1$ as an ...

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205 views

### 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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411 views

### 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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280 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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112 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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108 views

### 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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173 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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743 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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102 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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257 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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251 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

181 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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635 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

368 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 ...

**2**

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**1**answer

105 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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280 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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192 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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244 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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238 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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181 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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213 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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118 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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395 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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78 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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259 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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558 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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375 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., $P=\Bbb Q$ ...