  We study the dependence of geometric invariant theory quotients on the choice
of a linearization. We show that, in good cases, two such quotients are related
by a flip in the sense of Mori, and explain the relationship with the minimal
model programme. Moreover, we express the flip as the blow-up and blow-down of
specific ideal sheaves, leading, under certain hypotheses, to a quite explicit
description of the flip. We apply these ideas to various familiar moduli
problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos-
Wentworth, and the author. Along the way we display a chamber structure,
following Duistermaat-Heckman, on the space of all linearizations. We also give
a new, easy proof of the Bialynicki-Birula decomposition theorem.
