I have to prove the following:
Let $ K = \overline{K} $, consider $ X \subseteq \mathbb{P}^n $ irreducible projective variety and $ f \in \mathbb{K} [x_0, ..., x_n] $ homogenous polynomial of degree $ d $. Prove that $ X-Z(f) $ is isomorphic to an affine variety.
My doubt is:
How to prove this result?
Is not it necessary to ask that $ X \cap Z (f) \neq \emptyset $?
My idea was to use the Veronese application to see $ f $ as a linear function in a $ \mathbb{P}^N $ and so $ \mathbb{P}^N - Z (f) $ is an open affine.