Questions tagged [projective-geometry]

Question 1

I have a drawing that has a sloping line and a point not on that line. I want to draw a line passing through the given point that intersects the given line in a point outside of the paper. I believe ...

Question 2

I am reading the Gortz, Wedhorn's Algebraic Geometry, proof on Theorem 14.132 and stuck at some statements. ( Can anyone who have the Gortz, Wedhorn's book help? ) EDIT : This post is not duplicate. ...

Question 3

Let a Steiner conic $D$ be obtained from two pencils of lines $\check{A}$ and $\check{B}$, between which, as lines in $\check{\mathbb{P}^2}$, a projective map is given $f:\check{A} \to \check{B}$. Let ...

Question 4

Let $DBC$ be a triangle and $A'$ be a point inside the triangle such that $\angle DBA'$ is equal to $\angle A'CD$. Let $E$ be such that $BA'CE$ is a parallelogram. Show that $\angle BDE$ is equal to $\...

Question 5

I am an undergraduate math major who likes to draw, and I would like to learn the math behind perspective drawing. I recently watched this video: Everything about Perspective & Correct ...

Question 6

This is a problem I found on the Rick Miranda's book. Problem What is the minimum integer $k$ such that for every curve $X$ of a fixed genus $g$ there is a holomorphic map $F: X \rightarrow \mathbb{P}^...

Question 7

The game "Dobble" ("Spot it!" in the USA; see wikipedia for some details) is a card game. Each card has 8 symbols printed on them; each pair of card has exactly one common symbol. ...

Question 8

Body: In the usual field of real or complex numbers, division by 0 is undefined because no x satisfies 0x=1. However, various extensions (projective arithmetic, wheels, non-standard analysis, or ...

Question 9

The principle of duality in projective geometry, from wikipedia: In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by ...

Question 10

On page 144 of Coxeter's $\textit{Geometry Revisited}$, in the chapter on Projective Geometry, in the course of extending the Euclidean plane to the projective plane, Coxeter has this paragraph: "...

Question 11

Let $C$ be a nondegenerate conic in $\mathbb{P}^2$, and fix a line $t$. For each point $P\in C$, let $\ell_P$ denote the tangent to $C$ at $P$. Define the map $$ \Phi_C:\; P \longmapsto \ell_P\cap t. $...

Question 12

Let $A\in GL(3,\mathbb R)$ and let $[A]\in PGL(3,\mathbb R)$ be its projective class. Assume $[A]$ is real in $PGL(3,\mathbb R)$, i.e. $[A]$ is conjugate to its inverse $[A^{-1}]$. Geometrically (via ...

Question 13

Let $$ Q = \{[X:Y:Z:W]\in \mathbb{RP}^3 \mid XW - YZ = 0\}, $$ a smooth quadric surface. Using the Segre embedding $$ \Sigma:\ \mathbb{P}^1\times \mathbb{P}^1 \to \mathbb{P}^3,\quad ([x:y],[z:w]) \...

Question 14

The automorphism group of a smooth conic in $\mathbb{RP}^2$ is isomorphic to $\mathrm{PGL}(2,\mathbb{R})$. We get an embedding $$ \mathrm{PGL}(2,\mathbb{R}) \;\longrightarrow\; \mathrm{PGL}(3,\mathbb{...

Question 15

Hartshorne has the following theorem: Theorem 7.2. (Projective Dimension Theorem) Let $Y$, $Z$ be varieties of dimensions $r$, $s$ in $\mathbb{P}^n$. Then every irreducible component of $Y\cap Z$ has ...

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Questions tagged [projective-geometry]

I have a drawing where I have a sloping line and a point not on that line. How can I draw a line through that point that meets the line. [closed]

Proof of embeddibility of projective smooth $k$-scheme with dimension $d$ in $\mathbb{P}^{2d+1}_k$ ( Part 2, Gortz, Wedhorn )

What is the conic $D$ in this case?

Parallelogram with an external point connected to vertices

Mathematics for Perspective Drawing

Minimum degree of a holomorphic map from an algebraic curve of genus g to the Riemann sphere

How to sort the cards in Dobble/Spot it!

Can we define division by 0 in an extended algebraic structure? [duplicate]

Dual statements in Projective Geometry: Why sometimes two points becomes one line?

Proof that point-tangent pairs on circles reciprocate into tangent-point pairs on conics

A rational map from a conic to a line through its tangent intersection

In $PGL(n,\mathbb{R})$, is every real element strongly real?

Intersection of stabilizers in $PGL(3)$ of a smooth doubly ruled quadric surface in $\mathbb{P}^3$ and two skew lines on it

All all embeddings of $\mathrm{PGL}(2,\mathbb{R})$ into $\mathrm{PGL}(3,\mathbb{R})$ stabilizer of some smooth conic?

Hartshorne's proof of Theorem 7.2, the Projective Dimension Theorem.

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