Definition 101.4.1. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks.
[1] The letters DM stand for Deligne-Mumford. If f is DM then given any scheme T and any morphism T \to \mathcal{Y} the fibre product \mathcal{X}_ T = \mathcal{X} \times _\mathcal {Y} T is an algebraic stack over T whose diagonal is unramified, i.e., \mathcal{X}_ T is DM. This implies \mathcal{X}_ T is a Deligne-Mumford stack, see Theorem 101.21.6. In other words a DM morphism is one whose “fibres” are Deligne-Mumford stacks. This hopefully at least motivates the terminology.
[2] If f is quasi-DM, then the “fibres” \mathcal{X}_ T of \mathcal{X} \to \mathcal{Y} are quasi-DM. An algebraic stack \mathcal{X} is quasi-DM exactly if there exists a scheme U and a surjective flat morphism U \to \mathcal{X} of finite presentation which is locally quasi-finite, see Theorem 101.21.3. Note the similarity to being Deligne-Mumford, which is defined in terms of having an étale covering by a scheme.
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