Notation

Let \(F\) be \(\mathbb Q\), \(\mathbb R\), \(\mathbb C\), or any other field (if you know what that means, no problem if not!). Recall that \(\mathbb P^2(F)\) is defined as follows. We consider \(F^3\setminus \{(0,0,0)\}\) and define

if there exists \(\lambda \in F^\times \) such that \((x',y',z')=\lambda (x,y,z)\). The equivalence class of \((x,y,z)\) is denoted by

The points of \(\mathbb P^2(F)\) are the equivalence classes \([x:y:z]\). The lines of \(\mathbb P^2(F)\) are subsets of the form

where \((a,b,c)\neq (0,0,0)\). Incidence is the usual relation of belonging.

Problem: \(\mathbb P^2(F)\) as the projective completion of \(F^2\)

The goal of this problem is to prove that \(\mathbb P^2(F)\) is isomorphic to the projective completion of the affine plane \(F^2\).

Part 1: Homogeneous coordinates

(a)

Show that \(\sim \) is an equivalence relation on \(F^3\setminus \{(0,0,0)\}\).

Hint: Check reflexivity, symmetry, and transitivity.

(b)

For each of the following, answer YES or NO. Briefly justify your answer.

(i)

Is \([1:0:0]=[2:0:0]\) in \(\mathbb P^2(\mathbb Q)\)?

(ii)

Is \([1:0:0]=[1:1:1]\) in \(\mathbb P^2(\mathbb Q)\)?

(iii)

Is \([1:2:3]=[2:4:6]\) in \(\mathbb P^2(\mathbb Q)\)?

(iv)

Is \([1:2:3]=[2:4:5]\) in \(\mathbb P^2(\mathbb Q)\)?

(v)

Is \([1:1:0]=[2:2:0]\) in \(\mathbb P^2(\mathbb Q)\)?

(c)

Let \([x:y:z]\in \mathbb P^2(F)\) with \(z\neq 0\). Show that

\[ [x:y:z]=\left [\frac {x}{z}:\frac {y}{z}:1\right ]. \]

Part 2: Lines in \(\mathbb P^2(F)\)

(a)

Show that the expression

\[ ax+by+cz=0 \]

defines a well-defined condition on projective points. In other words, show that if \([x:y:z]=[x':y':z']\), then \(ax+by+cz=0\) if and only if \(ax'+by'+cz'=0\).

(b)

Describe the projective line passing through \([1:0:0]\) and \([0:0:1]\).

Hint: Find a nonzero triple \((a,b,c)\) such that both points satisfy \(ax+by+cz=0\). There will be many solutions; a one-dimensional space of solutions to be more precise.

(c)

Describe the projective line passing through \([1:0:0]\) and \([0:1:0]\).

(d)

Let \(a,b,c\in F\), with \((a,b)\neq (0,0)\), and consider the affine line

\[ \ell =\{(x,y)\in F^2: ax+by+c=0\}. \]

We define its projective closure as

\[ \overline {\ell }:=\{[x:y:z]\in \mathbb P^2(F): ax+by+cz=0\}. \]

Show that \(\overline {\ell }\) meets the line at infinity \(z=0\) in exactly one point, namely

\[ [-b:a:0]. \]

Hint: Substitute \(z=0\) into \(ax+by+cz=0\).

Part 3: The map on points

Let \(\mathcal A=(F^2,\mathcal L_{F^2},I_{F^2})\) be the affine plane over \(F\), and let \(\mathcal A^\ast =(\mathcal P^\ast ,\mathcal L^\ast ,I^\ast )\) be its projective completion.

Recall that the points of \(\mathcal A^\ast \) are of two types: ordinary points \((x,y)\in F^2\), and points at infinity \([\ell ]\), where \(\ell \) is a parallel class of affine lines.

Define a map \(f:\mathcal P^\ast \to \mathbb P^2(F)\) as follows. If \((x,y)\in F^2\), define \(f(x,y)=[x:y:1]\). If \([\ell ]\) is a point at infinity, choose a representative \(\ell \) with equation \(ax+by+c=0\), and define

To help visualize homogeneous coordinates and the projective completion of \(F^2\), you can click here for the Desmos 3D animation introduced in class.

(a)

Show that \(f\) is well-defined on points at infinity. In other words, show that if two affine lines are parallel, then they determine the same projective point at infinity.

Hint: Parallel affine lines have equations \(ax+by+c=0\) and \(\lambda ax+\lambda by+d=0\), for some \(\lambda \in F^\times \).

(b)

Show that \(f\) is injective.

Hint: Treat separately the cases of two ordinary points, one ordinary point and one point at infinity, and two points at infinity.

(c)

Show that \(f\) is surjective.

Hint: If \([x:y:z]\) has \(z\neq 0\), scale so that \(z=1\). If \(z=0\), show that \([x:y:0]=[-b:a:0]\) for some \((a,b)\neq (0,0)\).

Part 4: The map on lines

Recall that the lines of \(\mathcal A^\ast \) are of two types: ordinary affine lines \(\ell \in \mathcal L_{F^2}\), and the line at infinity \(\ell _\infty \).

Define a map \(g:\mathcal L^\ast \to \mathcal L_{\mathbb P^2(F)}\) as follows. If \(\ell \subseteq F^2\) is the affine line \(ax+by+c=0\), define

For the line at infinity, define

(a)

Show that \(g\) is injective.

Hint: If two affine lines have the same projective closure, compare their affine points with \(z=1\). Also, no affine line maps to \(z=0\).

(b)

Show that \(g\) is surjective.

Hint: Let \(L\) be a projective line \(Ax+By+Cz=0\). If \((A,B)\neq (0,0)\), show that \(L\) is the image of the affine line \(Ax+By+C=0\). If \((A,B)=(0,0)\), what is \(L\)?

Part 5: Incidence

Finally, show that the maps \(f\) and \(g\) preserve incidence. In other words, show that for every point \(p\in \mathcal P^\ast \) and every line \(\ell \in \mathcal L^\ast \),

(a)

Check this when \(p=(x,y)\) is an ordinary affine point and \(\ell \) is an ordinary affine line.

(b)

Check this when \(p=[\ell ]\) is a point at infinity and \(m\) is an ordinary affine line.

Hint: Write \(\ell \) as \(ax+by+c=0\) and \(m\) as \(a'x+b'y+c'=0\). The point at infinity \([\ell ]\) is incident with \(m\) exactly when \(m\) is parallel to \(\ell \).

(c)

Check this when \(p=[\ell ]\) is a point at infinity and the line is \(\ell _\infty \).

(d)

Conclude that \(\mathcal A^\ast \) and \(\mathbb P^2(F)\) are isomorphic as incidence geometries.