Could someone explain to me please, why is, in the case of a subset $ X_{ f_{d}} $ of $ \mathbb{P}^n ( \mathbb{C} ) $ défined by $ X_{f_{d}} = \{ (x_0 : \dots : x_n ) \in \mathbb{P}^n ( \mathbb{C} ) \ | \ f_d ( x_0 , ... , x_n ) = 0 \ \} $ such that, $ f_d $ is an homogeneous polynomial of degree $ d $ : $ X_{f_{d}} $ is a complex variety, if and only if :
$$ \dfrac{ \partial f_{d} }{ \partial x_{0} } (x_0 , \dots , x_n ) = \dots = \dfrac{ \partial f_{d} }{ \partial x_{n} } (x_0 , \dots , x_n ) = 0 \ \ \Longrightarrow \ \ x_0 = \dots = x_n = 0 $$
?
Thanks in advance for your help.