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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85 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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28 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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55 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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120 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
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116 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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70 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 ...
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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 ...
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87 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 ...
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130 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 ...
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94 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 ...
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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 ...
4
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210 views

Hirzebruch surfaces

How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Is it true that $F_{2}$ is the total space of a bundle with fibre SO(3) over $\mathbb{R}_{+}$?
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52 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.
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38 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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182 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 ...
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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, ...
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235 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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140 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 ...
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141 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 ...
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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 ...
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40 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 ...
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46 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 ...
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31 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 ...
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15 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 ...
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89 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?
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58 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 ...
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152 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 ...
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41 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 ...
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68 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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83 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 ...
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301 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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99 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 ...
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58 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 ...
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71 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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80 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 $\pi:X\rightarrow\mathbb{P}^1$ ...
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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 ...
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160 views

Complex projective space CP3 (Twistor space), as bundle space with base CP1, and fiber 4-D Minkowski space-time?

Twistor space, as complex projective space $CP3$, is related to Minkowski 4-D space time (metric $1, -1,-1,-1)$, by the incidence relation. Let $Z = (v_a, u^{\dot{a}})$ a point in $CP3$, where $v_a$ ...
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136 views

Intersection of a cone $x^2+y^2-z^2$ and a generic plane in $\mathbb{RP}^3$

If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$. The result will be a conic, but ...
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32 views

Atlas on the Grassmannian Variety

Let $G(k,n)$ the set of all $k$-dimensional sub-spaces of a vector complex space $V$ of dimension $n$. I know that it is possible to define the grassmannian as the quotient of $\chi(n,k)$ by $GL(k)$ ...
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$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the ...
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25 views

Is the action of the matrix UU^(t) always a projection? What can I say about I - UU^t?

The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal ...
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31 views

Generalization of Euler theorem for homogeneous polynomials

Euler's theorem for homogeneous polynomials is well known. If $F:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is a homogeneous polynomial, then we have: $x_{1}\frac{\partial F }{\partial x_{1}} + ... + ...
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13 views

Calculate the plane angle from 2D plane

I am analysing a squared plane from a perfect cube. This plane is distorted by the perspective view of a camera. I would like to know ask please, some approaches of how could I get to know the ...
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59 views

Is projective space really a moduli space for lines through the origin?

The Wikipedia page for Moduli spaces states that real projective space $\mathbb{RP}^n$ is a moduli space which parametrizes the space of lines in $\mathbb R^{n+1}$ passing through the origin. ...
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21 views

Vanishing points from three collinear points

I would like to find the 2D vanishing point from a three collinear points as is shown in "Multiple View Geometry in Computer Vision" Example 2.19 (see here). What I did so far: 1 - I've extracted ...
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26 views

Connection between Chladni Plates and Projective Geometry?

Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
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30 views

Construct sum of points on projective line

In a course in Geometry we where asked to geometrically construct the sum of two points $x$ and $y$ on a projective line by help of "the theorem on complete quadrilaterals". This theorem (as stated in ...