# Alternate Compactifications of Hurwitz Spaces

 Title: Alternate Compactifications of Hurwitz Spaces Author: Deopurkar, Anand Citation: Deopurkar, Anand. 2012. Alternate Compactifications of Hurwitz Spaces. Doctoral dissertation, Harvard University. Full Text & Related Files: Deopurkar_gsas.harvard_0084L_10308.pdf (1.066Mb; PDF) Abstract: We construct several modular compactifications of the Hurwitz space $$H^d_{g/h}$$ of genus g curves expressed as d-sheeted, simply branched covers of genus h curves. They are obtained by allowing the branch points of the cover to collide to a variable extent, generalizing the spaces of twisted admissible covers of Abramovich, Corti, and Vistoli. The resulting spaces are very well-behaved if d is small or if relatively few collisions are allowed. In particular, for d = 2 and 3, they are always well-behaved. For d = 2, we recover the spaces of hyperelliptic curves of Fedorchuk. For d = 3, we obtain new birational models of the space of triple covers. We describe in detail the birational geometry of the spaces of triple covers of $$P^1$$ with a marked fiber. In this case, we obtain a sequence of birational models that begins with the space of marked (twisted) admissible covers and proceeds through the following transformations: (1) sequential contractions of the boundary divisors, (2) contraction of the hyperelliptic divisor, (3) sequential flips of the higher Maroni loci, (4) contraction of the Maroni divisor (for even g). The sequence culminates in a Fano variety in the case of even g, which we describe explicitly, and a variety fibered over $$P^1$$ with Fano fibers in the case of odd g. Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10086270 Downloads of this work: