A point $\in \mathbb{P}^2(\mathbb{C})$

Question

What is the property that should be satisfied so that a point $\in \mathbb{P}^2(\mathbb{C})$ ? I am looking at an exercise where I have to find the flexes of a curve and this information is needed.

Answer 1

Recall that $\mathbb{P}^2(\mathbb{C}) = (\mathbb{C}^3\setminus\{0\})/\sim$ where $(z_1, z_2, z_3) \sim (w_1, w_2, w_3)$ if there is $\lambda \in \mathbb{C}\setminus\{0\}$ such that $(z_1, z_2, z_3) = \lambda(w_1, w_2, w_3)$.

Usually one writes an element of $\mathbb{P}^2(\mathbb{C})$ using homogeneous coordinates. That is, we write a point in $\mathbb{P}^2(\mathbb{C})$ as $[z_0, z_1, z_2]$; this denotes the image of $(z_0, z_1, z_2)$ under the projection map $\mathbb{C}^3\setminus\{0\} \to \mathbb{P}^2(\mathbb{C})$. As such, $[\lambda z_0, \lambda z_1, \lambda z_2] = [z_0, z_1, z_2]$ for every $\lambda \in \mathbb{C}\setminus\{0\}$.