# On the geometry of Deligne-Mumford stacks

Kresch, A (2009). On the geometry of Deligne-Mumford stacks. In: Abramovich, D; Bertram, A; Katzarkov, L; Pandharipande, R; Thaddeus, M. Algebraic Geometry: Seattle 2005. Providence, Rhode Island: American Mathematical Society, 259-271.

## Abstract

General structure results about Deligne–Mumford stacks are summarized, applicable to stacks of finite type over a field. When the base field has characteristic 0, a class of “(quasi-)projective” Deligne–Mumford stacks is identified,
defined to be those that embed as a (locally) closed substack of a smooth proper Deligne–Mumford stack having projective coarse moduli space. These conditions are shown to be equivalent to some well-studied hypotheses.

## Abstract

General structure results about Deligne–Mumford stacks are summarized, applicable to stacks of finite type over a field. When the base field has characteristic 0, a class of “(quasi-)projective” Deligne–Mumford stacks is identified,
defined to be those that embed as a (locally) closed substack of a smooth proper Deligne–Mumford stack having projective coarse moduli space. These conditions are shown to be equivalent to some well-studied hypotheses.