Alternate compactifications of Hurwitz spaces
Anand Deopurkar, Harvard
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 $H^d_g$, =
which parametrizes d sheeted, simply branched, genus g covers of the =
projective line. These compactifications are constructed by allowing =
degenererations where the branch points can collide in a prescribed way, =
recovering as a particular case the standard compactification by =
admissible covers.
After the general construction, I will focus on the case of $d =3D 3$. =
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.