Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

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26
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5answers
3k views

Help understanding Algebraic Geometry

I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with ...
25
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4answers
868 views

Geometry or topology behind the “impossible staircase”

This question on the topology of Escher games reminded me of a question I've had in my head for a little while now. Is there anything interesting geometric or topological that can be said about the ...
14
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1answer
205 views

Can a continuous map $S^2 \rightarrow S^2$ preserve orthogonality without being an isometry?

Suppose I have a map $\phi: S^2 \rightarrow S^2$ and I know that a) $\phi$ is continuous and bijective b) If $a$ and $b$ subtend an angle of $\pi / 2$ at the center of the sphere, then so do ...
14
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1answer
396 views

topologies of spaces in escher games

There have been a couple of games released (or in development) in the past couple of years which do some weird topological tricks: Echochrome (video), Crush (video), and Fez (video). Do the spaces ...
12
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3answers
4k views

Finding the Transform matrix from 4 projected points (with Javascript)

I'm working on a project using Chrome - JS and Webkit 3D CSS3 transform matrix. The final goal is to create a tool for artistic projects using projectors and animation - somewhat far away from using ...
12
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2answers
525 views

How should I think about very ample sheaves?

Definition. [Hartshorne] If $X$ is any scheme over $Y$, an invertible sheaf $\mathcal{L}$ is very ample relative to $Y$, if there is an imersion $i:X \to \mathbb{P}_Y^r$ for some $r$ such that ...
12
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1answer
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 ...
11
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1answer
1k views

Plücker Relations

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of decomposable ...
10
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2answers
235 views

Is a divisor in the hyperplane class necessarily a hyperplane divisor?

Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$. ...
10
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3answers
437 views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
10
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1answer
85 views

History of the point at infinity?

I'm curious to learn more about the history of the introduction of the concept of the point at infinity into mathematics. The sum of my knowledge of the historical aspect is from this paragraph (which ...
9
votes
4answers
898 views

Textbook for Projective Geometry

So while not actually a specific problem I'm struggling with, I was hoping for some of your insight! For a course, I'm currently reading Stillwell's Four Pillars of Geometry. While it does a nice ...
9
votes
2answers
788 views

Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
9
votes
1answer
311 views

A modern textbook on affine and projective spaces

Requirements: Scalar fields other than $\mathbb{R}$ and $\mathbb{C}$. Precise. Visual explanations are good, but they must complement definitions and proofs, not replace them. No repetition of text. ...
8
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1answer
300 views

Learning projective geometry

My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
7
votes
2answers
305 views

Is there a slick way to show that finite projective planes of $7$ points are unique up to isomorphism?

I was reading about the Fano plane, the smallest possible projective plane. After playing around with it, it seems that any projective plane of 7 points will be isomorphic to the Fano plane. ...
7
votes
2answers
285 views

How should I think of lines and planes in projective space?

I have been learning about projective varieties recently and I realised that I have some trouble trying to grasp what lines and planes are even in say $\Bbb{P}^3$. For one, how should I think about a ...
7
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1answer
465 views

Rectify image from congruent planar shape objects

I am implementing an algorithm to remove projective distortions on the following image. I understand this is possible by applying the following transformation: $$ \begin{matrix} 1 ...
7
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2answers
307 views

Perspective problem - trapezium turned square

True or false: If you draw a trapezium on the ground, there always exists a point above (but not necessarily directly above) the trapezium such that the trapezium looks like a square from that point. ...
6
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3answers
5k views

The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
6
votes
2answers
132 views

Construction of the projective plane over $\mathbb{F}_3$

I have a question about constructing projective plane over $\mathbb{F}_3$. We first establish seven equivalence classes $P= \{ [0,0,1], [0,1,0], [1,0,0], [0,1,1], [1,1,0], [1,0,1], [1,1,1] \}$. ...
6
votes
1answer
248 views

Is the graph of morphism of projective varieties $X \rightarrow Y$ closed in $X \times Y$?

The graph of a morphism $X \rightarrow Y$ is closed in $X \times Y$ if $X$ and $Y$ are affine varieties. What if $X$ and $Y$ are projective varieties? I am still not quite familiar with projective ...
6
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1answer
122 views

Why is $\mathbb{R} P^n$ called projective space?

I know that: If one defines an equivalence relation on $\mathbb{R}^{n+1}-\{0\}$ by $$x\sim y \iff y=tx$$ for some nonzero real number $t$, where $x,y\in\mathbb{R}^{n+1}-\{0\}$, Then The real ...
6
votes
0answers
376 views

Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I ...
5
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1answer
146 views

If a line bundle and its dual both have a section (on a projective variety) does this imply that the bundle is trivial?

Is there any reason that, on a projective variety X, if a line bundle L has a (non-zero) section and also its dual has a section then this implies that L is the trivial line bundle?
5
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1answer
77 views

Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
5
votes
3answers
110 views

Understanding cross ratio and harmonic conjugates

I'm studying projective geometry and I'm really having trouble with ''grokking'' what's it all about. Is there an easy/intuitive/visual way to understand cross ratio? I understand that it's ...
5
votes
3answers
307 views

Projective spaces with Zariski topology

Why $\mathbb{P}^1\times\mathbb{P}^1\not\cong\mathbb{P}^2$ where the projective spaces have the Zariski topology?
5
votes
1answer
99 views

projective cubic

I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set ...
5
votes
1answer
47 views

Finding the inverse of a map from $CP^1$ to $S^2$

Given the map: $$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ How would I go about ...
5
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2answers
198 views

Implicitization of a cubic

Let $\nu$ be the map $\mathbb{P}^1 \to \mathbb{P}^2$ defined by $$\nu([X_0,X_1])=[A_0(X_0,X_1),A_1(X_0,X_1),A_2(X_0,X_1)]$$ where the $(A_i)$ are homogenous polynomials of degree 3, without common ...
5
votes
2answers
371 views

Analytically flavored book in projective geometry

I am looking for a book in projective geometry, using the apparatus of linear algebra, complex analysis, and, perhaps, modern algebra, in full. The counterexample is the Hartshorne's book on ...
5
votes
1answer
228 views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus ...
5
votes
1answer
230 views

Problem in Deducing Perspective Projection Matrix

I understand the traditional way(use similar triangle and make depth value linear) to deduce the perspective projection matrix. But I want to try another approach after I read this text: Fundamentals ...
5
votes
1answer
245 views

Parametrizing a conic in projective space

I am just beginning to learn algebraic geometry. An exercise in Reid, p. 24 is to prove that if $Q(x,y,z)$ is a quadratic form over a field $k$ with at least 4 elements, and $Q$ vanishes on the zero ...
5
votes
0answers
87 views

How to calculate the dimension of the intersection of projection of varieties?

In $\mathbb P_{n+1}$ we consider $d$ varieties $V_1,\ldots V_d$, each of them is defined by $d-1$ equations, as follows: we have $d-1$ polynomials $f_2(X_0,\ldots,X_n),\ldots,f_d(X_0,\ldots,X_n)$ and ...
5
votes
1answer
212 views

Möbius transforms on the projective line

Let's suppose that I have a mapping $$ \phi:\mathbb{P}^1\rightarrow\mathbb{P}^1 $$ of degree two. How can I use Möbius transforms to write this map as $$ \phi([x:y])=\frac{y^2}{x^2}? $$ I already ...
5
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0answers
125 views

Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$

I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$. Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines. There are $q+1$ ...
4
votes
3answers
199 views

what is the definition of a line in $\mathbb{P}^n(k)$ + how to compute the hilbert polynomial of two intersecting lines?

(1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic ...
4
votes
1answer
98 views

Difference between the real projective plane and the complex projective plane

Well the title says it all. If we consider the $P^2(\Bbb R)$ and the $P^2(\Bbb C)$, and we compare them, my guess is that it will be like a round $\Bbb R^2$ versus a sphere. I don't have very good ...
4
votes
1answer
74 views

Description of varieties in $\mathbb{P}^2\times \mathbb{P}^1$

If $[x:y]$ are coordinates of $\mathbb{P}^1$ and $[X:Y:Z]$ are coordinates of $\mathbb{P}^2$, what do the following varieties look like? $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$ ...
4
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1answer
184 views

How to show $\mu(\overline{\mathbb{R}}) = \overline{\mathbb{R}}$ iff $a, b, c, d \in \mathbb{R}$ for $\mu(z) = \frac{az+b}{cz+d}$?

Let $\mu(z) = \frac{az+b}{cz+d}$ be a Möbius transformation. I want to show that $$\mu(\overline{\mathbb{R}}) = \overline{\mathbb{R}} \iff a, b, c, d \in \mathbb{R}.$$ What would be an elegant, and ...
4
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1answer
169 views

Proof of Moisezon Theorem

We call a compact complex manifold Moisezon manifold, if its dimension coincides with the algebraic dimension, i.e. it has as many algebraically independent meromorphical functions as its complex ...
4
votes
1answer
76 views

Isomorphisms of $\mathbb P^1$

Prove that every isomorphism of $\mathbb P^1$ (over an algebrically closed field $\mathbb K$) is of the form $$ \phi(x_0: x_1) = (ax_0+bx_1 : cx_0 + dx_1) $$ where $\begin{pmatrix} a & b ...
4
votes
1answer
112 views

Dual projective surface

Let $X \subset \mathbb{P}^3$ be a smooth surface of degree $d>1$ and consider the Gauss map $X \to \mathbb{P}^{3*}$, which sends a point of $X$ to its tangent plane. To see that the image $X^*$ of ...
4
votes
1answer
249 views

Pullback of very ample sheaf again very ample? And other questions.

Let $S \subseteq \mathbb{P}^n$ be a smooth projective surface with given embedding in projective space. Moreover, let $X$ be another smooth surface and let there be a map $\pi: X \rightarrow S$ that ...
4
votes
1answer
160 views

intrinsic and geometric definition of blow-up

Suppose I am given an algebraic variety $X$ and a (closed) point $x \in X$. I know of two descriptions of the blow-up of $X$ at $x$. One is intrinsic but not geometric: if $\mathcal{I}_x$ denotes the ...
4
votes
1answer
144 views

How to show that the image of a certain projective embedding is an algebraic curve?

I found the following claim in a paper by Griffiths and Harris : Start with a complex torus $\mathbb{C}/\Lambda$. The vector space of meromorphic functions having period lattice $\Lambda$ and a pole ...
4
votes
1answer
36 views

projective plane over algebraic structure

I know that it is possible to build a projective plane over any field. I also know that it is possible to build a projective plane over Hamilton's quaternions. My question is: Is it possible to build ...
4
votes
1answer
81 views

some fun with holomorphic line bundles

These are probably trivial questions... (for the experts) I'd like to get convinced (perhaps an intuitive/geometric explanation will be more effective than a formal one) of the following facts: i. ...