Abstract

Ample line bundles on schemes have many equivalent characterizations, but one key feature is that they can be used to embed the base scheme into a projective space. I will present two natural definitions of an ample vector bundle on a stack, not clearly equivalent, one generalizing a cohomological property and the other generalizing the embedding property of ample line bundles. Using the latter, and extending work of Kresch, I will explain how an ample vector bundle on a tame stack induces an embedding into [V^{s/GL_r] where V}s is the twisted affine GIT stable locus in some polynomial representation V of GL_r. An application is the construction of moduli of stacks with ample vector bundle and in particular a very general stack of tame stacky curves. This is joint with Daniel Bragg and Martin Olsson.