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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.

evinda
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  • The question as such only makes sense if $P^2$ is considered as a subset of some langer space. So you should let us know in which space you have $P^2$ as a subset. (and perhaps some more context) – user39082 Dec 12 '14 at 16:00

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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 $\lambda \in \mathbb{C}\setminus\{0\}$.

Michael Albanese
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  • Michael I want to find the solutions of $x^3+y^3=0$ in $\mathbb{P}^2(\mathbb{C})$ ? So, how could we find all the solutions in $\mathbb{P}^2(\mathbb{C})$ ? – evinda Dec 12 '14 at 16:59
  • You have to look for solutions in $C^3$. Their equivalence classes give you solutions in $P^2$. – user39082 Dec 12 '14 at 19:03
  • In this case, the solutions in $C^3$ are of the form $(s,-s,t)$, so solutions in $P^2$ are of the form $\left[1:-1:c\right]$ with $c\in C$. – user39082 Dec 12 '14 at 19:06