Projective Equivalence and Plane Conics

Projective Equivalence of Points

In projective geometry, the group $\mathrm{PGL}_{n+1}(k)$ acts transitively on certain configurations of points in $\mathbb{P}^n(k)$.

• $\mathbb{P}^1(k)$: 3-Transitivity

  Any ordered triple of distinct points in $\mathbb{P}^1(k)$ can be mapped to any other such triple by a unique projective transformation. That is:

  Given any three distinct points $(P_1, P_2, P_3)$ and $(Q_1, Q_2, Q_3)$ in $\mathbb{P}^1(k)$, there exists a unique $T \in \mathrm{PGL}_2(k)$ such that $T(P_i) = Q_i$ for $i = 1, 2, 3$.

  This property reflects the 3-dimensional nature of $\mathrm{PGL}_2(k)$ and the fact that homogeneous coordinates in $\mathbb{P}^1(k)$ are only defined up to scalar.

• $\mathbb{P}^2(k)$: 4-Transitivity in General Position

  Any ordered quadruple of points in $\mathbb{P}^2(k)$ in general position (i.e. no three lie on the same line) is projectively equivalent to any other such quadruple. More precisely:

  If $(P_1, P_2, P_3, P_4)$ and $(Q_1, Q_2, Q_3, Q_4)$ are two ordered sets of points in general position in $\mathbb{P}^2(k)$, then there exists a unique $T \in \mathrm{PGL}_3(k)$ such that $T(P_i) = Q_i$ for $i = 1, 2, 3, 4$.

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Conics in the Projective Plane

A conic in $\mathbb{P}^2(k)$ is defined by a homogeneous polynomial of degree 2:

\[f(x, y, z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz = 0\]

This can be represented compactly using a symmetric $3 \times 3$ matrix $M$:

\[f(x, y, z) = \begin{bmatrix} x & y & z \end{bmatrix} M \begin{bmatrix} x \\ y \\ z \end{bmatrix}\]

where

\[M = \begin{bmatrix}a & d & e \\ d & b & f \\ e & f & e\end{bmatrix}.\]

We call $M$ as a matrix representation of the conic.

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Types of Conics

• A smooth conic is one which the corresponding matrix is invertible.
• A degenerate conic is one which the corresponding matrix is degenerate. Geometrically, it is two lines (possibly overlaping)

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Theorem (5 Points Determine a Conic).
Let $\mathrm{char}(k) \ne 2$. Given five distinct points in $\mathbb{P}^2(k)$, no four of which lie on a line, there exists a unique conic passing through all five points.

Proof.
TODO

Theorem (Projective Equivalence of Conics).
Assuming $\mathrm{char}(k) \ne 2$. Any two smooth conics in $\mathbb{P}^2(k)$ are projectively equivalent. That is, there exists a projective transformation $T \in \mathrm{PGL}_3(k)$ mapping one conic onto the other.

In particular, every smooth conic is projectively equivalent to the standard conic $ x^2 + y^2 + z^2 = 0 $.

Proof.
TODO