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

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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 ...
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100 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
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31 views

Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
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91 views

Find the intersection of two lines entirely outside the given sheet of paper by straightedge alone

This is a problem from Courant:"Two straight lines entirely outside the given sheet of paper are each given by two pairs of straight lines intersecting at points of the lines outside the paper. ...
5
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59 views

Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
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60 views

Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
5
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138 views

Harris, Exercise 10.28 (weighted projective spaces)

So I recently started teaching myself about weighted projective spaces from Harris' Algebraic geometry. It was going well until I came across this exercise, which has me stumped: "Show that any ...
5
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123 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 ...
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46 views

Morphism between surfaces

Suppose that $S$ is a surface of general type. Let $K_S$ the canonical bundle of $S$ and $\phi=\phi_{K_S}$ the canonical map. Suppose that the canonical map is a morphism from $S$ to ...
4
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85 views

Bridging the gap between classical and modern projective geometry

The language of projective geometry is quite common in modern mathematics. For this reason I'd like to learn this subject, however, modern treatments are incredibly abstract. Now, I'm vaguely aware of ...
4
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77 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
4
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98 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
4
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95 views

Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
4
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44 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 ...
3
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29 views

About an example of normal bundle of a curve over a surface

I know the definition of the normal bundle $N_{C/S}$ of a curve $C$ over a surface $S$ as the cokernel of the injection $T_C \subset T_S|_C$ where $T$ is the tangent bundle. I would like to exhibit an ...
3
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42 views

Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
3
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115 views

Moduli space of algebraic surfaces Vs moduli space of curves

Define the surface $S$ as the complete intersection of four quadrics $Q_i$ with $i=1,2,3,4$ in $\mathbb{P}^6$ (complex six dimensional projective space) i.e. $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4$$. Put ...
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52 views

Weighted Projective Spaces and varieties

Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with ...
3
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41 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
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340 views

Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
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193 views

Irreducible closed subsets of projective varieties

I want to prove the following lemma: Let $X \subset \mathbb{P}^n$ be a projective variety. Let $W \subset X$ be a closed irreducible set. Then $W$ is also a projective variety. My idea is as ...
3
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50 views

Do we have homogeneous coordinates for probabilities?

As a roboticist, implementing visual odometry on a robot, homogeneous coordinates are convenient for projections of a non-moving object on an image sensor at $t$ and $t+1$ to estimate its position, ...
3
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245 views

There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
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143 views

Geometric interpretation of a certain property of graded rings

Let $k$ be a field, and $A$ a (commutative associative unital) $k$-algebra. It is known that there is a natural isomorphism between ‘regular’ (in the sense of regular maps of varieties) actions of ...
3
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142 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 ...
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89 views

Solving some 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 ...
3
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0answers
115 views

The geometry of $\operatorname{PSO}(4)$ and the quaternions

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 ...
2
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27 views

Inverting an isometric projection?

I'm trying to invert a function that takes points on a 2-d plane to an isometric projection of that plane. This function is encoded as follows (as part of the Isomer library): ...
2
votes
0answers
41 views

3D projection onto 2D plane to determine transformation matrix?

I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes: I basically have three vertices of a rigid ...
2
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0answers
42 views

Show that there exists a coordinate system in $\mathbb{P}^n$ such that $P_0=(1:0:\cdots:0),\ldots,P_n=(0:0:\cdots:1),P_{n+1}=(1:\cdots:1)$

Proposition: Let $P_0,\ldots,P_{n+1}$ be $n+2$ points in $\mathbb{P}^n$ such that every $n+1$ are in general position. There exists a coordinate system in $\mathbb{P}^n$ such that ...
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22 views

Generalization of a Result Concerning Projective Planes

Let $\mathcal P$ denote the set of all possible orders of projective planes. For $q\in\mathcal P$, let $PG_2(q)$ denote the projective plane of order $q$. There is a theorem due to James Singler ...
2
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51 views

Visualizing projective closures - is it okay to just think of the affine case?

This question is quite general and has been discussed on MSE before, however my case is a little bit different and I'm wondering about the geometric interpretation of a specific example. I think that ...
2
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0answers
34 views

Number of parameter of a quadric

Suppose for example that $S$ is an algebraic complex surface contained in $\mathbb{P}^6$. $S$ is the complete intersection of four quadrics in the six dimensional projective space. If i take a quadric ...
2
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20 views

Computing projective coordinates and equation of a plane.

The question We are given a vector space with basis $\{e_1,e_2,e_3\}$ and a projective frame, $\{ [e_1],[2e_1 + e_2],[e_2 - e_3] \}$ and a fixed point $[3e_1 + e_3]$. What is the coordinates for the ...
2
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17 views

Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
2
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0answers
37 views

A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically.

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically. A specific construction of a set of ...
2
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0answers
47 views

Uniqueness of a projective transformation

Just as there exists a unique projective transformation that takes three points in $\mathbb{CP}^1$ to three other points in $\mathbb{CP}^1$, how many points do I need for the corresponding question in ...
2
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0answers
61 views

Singular Conics and Intersection of Line with a Conic

I've been working through Silverman and Tate's book Rational Points on Elliptic Curves. They use conic equations as an introduction to singular/nonsingular curves. I've reproduced the problem with my ...
2
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0answers
22 views

Scaling axes to reflect perspective plane of an image.

Suppose I have an image which contains a road in it, and I want to be able to pinpoint the locations of cars on that road using pre-calibrated distances. How to do that using formulas that map pixel ...
2
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0answers
106 views

Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
2
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0answers
71 views

A question from “Foundations of Projective Geometry” by Hartshorne.

"Foundations of Projective Geometry" by Hartshorne says the following: The completion of the affine plane of four points is a projective plane with 7 points. The affine plane of $4$ points is ...
2
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0answers
160 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
2
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0answers
44 views

Projection from a point to a plane - confused about terminology.

Edit: It seems rude to delete the question, but I have my answer now thanks to rghthndsd. I'm a bit unsure about the terminology in a homework question I'm doing, and I can't find any clear answers ...
2
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0answers
69 views

Projectivizing a group: how to go from AGL(n,K) to PGL(n+1,K)

$ \newcommand{\GL}{\operatorname{GL}} \newcommand{\AGL}{\operatorname{AGL}} \newcommand{\PGL}{\operatorname{PGL}} $Given an irreducible matrix group $G_{\infty,0} \leq \GL(n,K)$, I form the group ...
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0answers
92 views

Projective Geometry Question.

I want to learn projective gemetry. I have the coxeter book and found some youtube videos. I have just started. I have tried answering this Challege question from the video, and so far. I have ...
2
votes
0answers
112 views

Finding the equations of a variety under a projection map

Suppose I have a projective variety $X$ in $\mathbb{P}^N$ ($N >2$, say) defined as the zero set of some homogeneous polynomials $f_1, \ldots, f_r$. Consider the projection map $[x_0: x_1 : \cdots ...
2
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0answers
59 views

Are there in $(\mathbb{C}[x,y,z]/(x^3+y^3+z^3))_{x}$ exactly $12$ lines?

Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ be the coordinate ring of the affine variety defined by the equation $x^3+y^3+z^3=0$. We can consider the localization in the element $x$, denoted by $R_x$. I ...
2
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0answers
76 views

Configuration analogues of projective spaces?

In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points. The Fano plane is a configuration, with 3 points on each line, and 3 ...
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82 views

How can I compute the multiplicities of the projection map over a smooth projective plane curve?

Suppose that $F$ is a non-singular homogeneous polynomial in $\mathbb{C}^3$ and let $X$ be its zero locus, which is well-defined in $\mathbb{P}^2$. Consider the function $\pi:X\rightarrow\mathbb{P}^1$ ...
2
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0answers
52 views

Isomorphism of $P(V)$ and $P(V^*)$

Let $V$ be a finite-dimensional left vector space over a division ring $K$, and let $V^*$ the dual right vector space (consisting of all linear functions from $V$ to $K$). We can (and will) treat ...