Girish Mahajan (Editor)

Deligne–Mumford stack

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In algebraic geometry, a Deligne–Mumford stack is a stack F such that

  • (i) the diagonal F → F × S F is representable (the base change to a scheme is a scheme), quasi-compact and separated.
  • (ii) There is a scheme U and étale surjective map U →F (called the atlas).

    • Deligne and Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne-Mumford stacks.

    If the "étale" is weakened to "smooth", then such a stack is called an Artin stack. An algebraic space is Deligne–Mumford.

    A key fact about a Deligne–Mumford stack F is that any X in F ( B ) , B quasi-compact, has only finitely many automorphisms.

    A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme.

    References

    Deligne–Mumford stack Wikipedia
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