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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.

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    $\begingroup$ Here is something for you to consider: what if the integers $a_0,\dots,a_n$ all equal a common integer $\ell>1$. Then your stack is a $\mu_\ell$-gerbe over $\mathbb{P}(a_0,\dots,a_n) \cong \mathbb{P}^n$. So no map $\phi$ from a scheme $Z$ to your stack is a closed embedding, even if it does satisfy the criterion you mention. $\endgroup$ Apr 5, 2023 at 11:27
  • $\begingroup$ @JasonStarr i agree, but that was for weights equal to 1. And when all weights are equal, the second question has that answer an easy answer. In general, I was just looking whether there are conditions on sections of tensor powers of a line bundle to give a closed embedding. $\endgroup$ Apr 5, 2023 at 19:00
  • $\begingroup$ What? There are no closed embeddings at all in that case. $\endgroup$ Apr 5, 2023 at 21:11
  • $\begingroup$ Am I misunderstanding something? The coarse moduli space is just ordinary $ \mathbb{P}^n $ in that case, no? $\endgroup$ Apr 5, 2023 at 22:36
  • $\begingroup$ There are no closed embeddings of a scheme into a $\mu_\ell$ gerbe. $\endgroup$ Apr 5, 2023 at 23:03

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