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Artin stack

• Example: the moduli stack of stable curves. Proving it is Artin uses a method of tricanonical embeddings, constructed using Hilbert schemes.
• geometric points do not usually admit universal deformation rings.
• Can prove Artin stacks are Deligne-Mumford stacks using an unramified diagonal argument.
• Common requirement: diagonals that are represented by separated algebraic space of finite type.