  We construct a sequence of modular compactifications of the space of marked
trigonal curves by allowing the branch points to coincide to a given extent.
Beginning with the standard admissible cover compactification, the sequence
first proceeds through contractions of the boundary divisors and then through
flips of the so-called Maroni strata, culminating in a Fano model for even
genera and a Fano fibration for odd genera. While the sequence of divisorial
contractions arises from a more general construction, the sequence of flips
uses the particular geometry of triple covers. We explicitly describe the Mori
chamber decomposition given by this sequence of flips.
