Let $ Z $ be a proper or even, projective scheme. I have two related questions. (everything is over complex numbers)
(1) What is a criteria for a map $ \phi: Z \rightarrow \left[ \mathbb{A}^{n+1} - (0) / \mathbb{G}_m \right] _{a_0, \cdots , a_n} $ to be a closed embedding? Here the object on the right is the weighted projective stack with weights $ a_0, \cdots , a_n $.
(2) What is a criteria for a map $ \psi: Z \rightarrow \mathbb{P}(a_0, \cdots, a_n) $ to be a closed embedding? Here the object on the right is the weighted projective scheme $ \operatorname{Proj} \mathbb{C} [x_0, \cdots , x_n] $ with intederminates $ x_i $ of degree $ a_i $ which is also the coarse moduli space of the stack above.
When all weights are equal to $ 1 $, the question of (1) atleast, should reduce to the criteria of sections of a globally generated line bundle separating points and separating tangents that everyone learns as a beginner.