Alternate compactifications of Hurwitz spaces
-- Anand Deopurkar, October 21, 2011

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Moduli spaces of geometrically interesting objects are usually
not compact. They need to be compactified by allowing certain
carefully chosen degenerations. Often, this can be done in
several ways, leading to different birational models that are
related in interesting ways. I will describe a range of
compactifications of the Hurwitz space H<sup>d</sup><sub>g</sub>, which
parametrizes d-sheeted, simply branched, genus g covers of
the projective line. These compactifications are constructed by
allowing degenerations where the branch points can collide in
a prescribed way, recovering as a particular case the standard
compactification by admissible covers.
   
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After the general construction, I will focus on the case d
= 3. In this case, the above construction gives a sequence of
compactifications which contract the boundary divisors in the
admissible cover compactification. I will construct a sequence
of yet more compactifications that modify the interior,
featuring the classical Maroni invariant of trigonal curves.