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Birational geometry of the space of rational curves in homogeneous varieties

In this thesis, we investigate the birational geometry of the space of rational curves in various homogeneous spaces, with a focus on the quasi-map compactification induced by the Quot and Hyperquot functors. We first study the birational geometry of the Quot scheme of sheaves on P1 via techniques from the Mori program, explicitly describing its associated cones of ample and effective divisors as well as the various Mori chambers within the latter. We compute the base loci of all effective divisors, and give a conjectural description of the induced birational models. We then partially extend our results to the Hyperquot scheme of sheaves on P', which gives the analogous compactification for rational curves in flag varieties. We fully describe the cone of ample divisors in all cases and the cone of effective divisors in certain ones, but only claim a partial description of the latter in general. Thesis Supervisor: James McKernan Title: Professor