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1. Deopurkar, Anand. Alternate Compactifications of Hurwitz Spaces.
Degree: PhD, Mathematics, 2012, Harvard University
We construct several modular compactifications of the Hurwitz space (Hdg/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 (P1) 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 (P1) with Fano fibers in the case of odd g.
MathematicsAdvisors/Committee Members: Harris, Joseph D. (advisor), Harris, Joseph (committee member), Mazur, Barry (committee member), Vakil, Ravi (committee member).
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APA (6th Edition):
Deopurkar, A. (2012). Alternate Compactifications of Hurwitz Spaces. (Doctoral Dissertation). Harvard University. Retrieved from http://nrs.harvard.edu/urn-3:HUL.InstRepos:10086270
Chicago Manual of Style (16th Edition):
Deopurkar, Anand. “Alternate Compactifications of Hurwitz Spaces.” 2012. Doctoral Dissertation, Harvard University. Accessed November 13, 2019. http://nrs.harvard.edu/urn-3:HUL.InstRepos:10086270.
MLA Handbook (7th Edition):
Deopurkar, Anand. “Alternate Compactifications of Hurwitz Spaces.” 2012. Web. 13 Nov 2019.
Deopurkar A. Alternate Compactifications of Hurwitz Spaces. [Internet] [Doctoral dissertation]. Harvard University; 2012. [cited 2019 Nov 13]. Available from: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10086270.
Council of Science Editors:
Deopurkar A. Alternate Compactifications of Hurwitz Spaces. [Doctoral Dissertation]. Harvard University; 2012. Available from: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10086270