 
Summary: GOOD MODULI SPACES FOR ARTIN STACKS
JAROD ALPER
Abstract. We develop the theory of associating moduli spaces with nice geometric properties to
arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks.
1. Introduction
1.1. Background. David Mumford developed geometric invariant theory (GIT) ([GIT]) as a means
to construct moduli spaces. Mumford used GIT to construct the moduli space of curves and rigid
ified abelian varieties. Since its introduction, GIT has been used widely in the construction of
other moduli spaces. For instance, GIT has been used by Seshadri ([Ses82]), Gieseker ([Gie77]),
Maruyama ([Mar77]), and Simpson ([Sim94]) to construct various moduli spaces of bundles and
sheaves over a variety as well as by Caporaso in [Cap94] to construct a compactification of the
universal Picard variety over the moduli space of stable curves. In addition to being a main tool in
moduli theory, GIT has had numerous applications throughout algebraic and symplectic geometry.
Mumford's geometric invariant theory attempts to construct moduli spaces (e.g., of curves) by
showing that the moduli space is a quotient of a bigger space parameterizing additional information
(e.g. a curve together with an embedding into a fixed projective space) by a reductive group. In
[GIT], Mumford systematically developed the theory for constructing quotients of schemes by
reductive groups. The property of reductivity is essential in both the construction of the quotient
and the geometric properties that the quotient inherits.
It might be argued though that the GIT approach to constructing moduli spaces is not entirely
