Deligne–Mumford stack
In algebraic geometry, a Deligne–Mumford stack is a stack that behaves, in many respects, like an algebraic variety or an orbifold, while still allowing mild stacky phenomena such as finite stabilizer groups. More precisely, a stack over schemes is Deligne–Mumford if its diagonal is sufficiently well behaved and if it admits an étale surjective cover by a scheme (an atlas).
Pierre Deligne and David Mumford introduced this notion in their 1969 paper on the irreducibility of the moduli space of algebraic curves, where they showed that the moduli stack of stable curves of fixed arithmetic genus is a proper smooth Deligne–Mumford stack over . Since then, Deligne–Mumford stacks have become a basic tool in moduli theory and in modern intersection theory, for instance in Gromov–Witten theory.
