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I am new to stacks and algebraic spaces. I have the following question: Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...

Let $G= \mathbb{G}_m^k$ act on a variety $X$. Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$. Does it ...