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The moduli space of stable quotients, introduced by Marian, Oprea, and Pandharipande, provides a nonsingular compactification of the moduli space of degree d maps from smooth genus 1 curves into projective space ℙⁿ. This is done by allowing the domain curve to have nodal singularities and by admitting certain rational maps. The rational maps are introduced in the following way: A map to projective space can be defined by a quotient bundle of the trivial bundle on the domain curve; in the compactification, the quotient bundle is replaced by a sheaf which may not be locally free. The boundary is filtered by the degree of the torsion subsheaf of the quotient. Yijun Shao has defined a similar compactification of the moduli space of degree d maps from ℙ¹ into a Grassmannian. A blow-up process is carried out on the compactification in order to produce a boundary which is a simple normal crossings divisor: The closed subschemes in the filtration of the boundary are blown up in order of decreasing torsion. In this thesis, we carry out an analogous blow-up process on the moduli space of stable quotients. We show that the end result is a nonsingular compactification which has as its boundary a simple normal crossings divisor.

Let Md be the moduli space of algebraic maps (morphisms) of degree d from P^1 to a fixed Grassmannian. The main purpose of this thesis is to provide an explicit construction of a compactification of Md satisfying the following property: the compactification is a smooth projective variety and the boundary is a simple normal crossing divisor. The main tool of the construction is blowing-up. We start with a smooth compactification given by Quot scheme, which we denote by Qd. The boundary Qd\Md is singular and of high codimension. Next, we give a filtration of the boundary Qd\Md by closed subschemes: Zd,0 subset Zd,1 subset ... Zd,d-1=Qd\Md. Then we blow up the Quot scheme Qd along these subschemes succesively, and prove that the final outcome is a compactification satisfying the desired properties. The proof is based on the key observation that each Zd,r has a smooth projective variety which maps birationally onto it. This smooth projective variety, denoted by Qd,r, is a relative Quot scheme over the Quot-scheme compactification Qr for Mr. The map from Qd,r to Zd,r is an isomorphism when restricted to the preimage of Zd,r\ Zd,r-1. With the help of the Qd,r's, one can show that the final outcome of the successive blowing-up is a smooth compactification whose boundary is a simple normal crossing divisor.

We compute the exact partition function of two dimensional N=(2,2) supersymmetric gauge theories on S². For theories with SU(2|1)_A invariance, the partition function admits two equivalent representations corresponding to localization on the Coulomb branch or the Higgs branch, which includes vortex and anti-vortex excitations at the poles. For SU(2|1)_B invariant gauge theories, the partition function is localized to the Higgs branch which is generically a Kähler quotient manifold. The resulting partition functions are invariant under the renormalization group flow. For gauge theories that flow in the infrared to Calabi-Yau nonlinear sigma models, the partition functions for the SU(2|1)_A (resp SU(2|1)_B) invariant theories compute the Kähler potential on the Kähler moduli (resp. complex structure moduli) of the Calabi-Yau manifold. We also compute the elliptic genus of such theories in the presence of Stückelberg fields and show that they are modular completions of mock Jacobi forms.

In this thesis, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes H(1,X(d),n) of morphisms of degree n from elliptic curves to the modular curve X(d), d&amp / #8804 / 3. Ultimately, we show that the moduli spaces, considered, are fiber spaces over the affine line A&sup1 / with fibers determined by the components of H (1,X(d),n).

A genus-two Riemann surface admits a canonical decomposition into Dirichlet polygons determined by its six Weierstrass points. All possible associated graphs are determined explicitly from circle packing problems, solved by systems of linear inequalities whose solutions determine a finite 6-dimensional polyhedral complex in 12-dimensional space. The 6-dimensional Moduli Space of genus-two Riemann surfaces inherits a canonical explicit decomposition into Euclidean polyhedra, giving new natural coordinates for the Teichmuller Space of all possible constant curvature geometries on a marked genus-two surface.

Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2007. / Typescript. Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (doctoral)--Bonn, 1997. / Includes bibliographical references (p. 93-95).

Thesis (doctoral)--Bonn, 1997. / Vita. Includes bibliographical references (p. 167-168).