The following is my M.Sc. thesis on moduli space techniques in algebraic and symplectic geometry. It is divided into the following two parts: the rst part is devoted to presenting moduli problems in algebraic geometry using a modern perspective, via the language of stacks and the second part is devoted to studying moduli problems from the perspective of symplectic geometry. The key motivation to the rst part is to present the theorem of Keel and Mori [20] which answers the classical question of under what circumstances a quotient exists for the action of an algebraic group G acting on a scheme X. Part two of the thesis is a more elaborate description of the topics found in Chapter 8 of [28].

Barycentric coordinates are functions on a polygon, one for each vertex, whose values are coefficients that provide an expression of a point of the polygon as a convex combination of the vertices. Wachspress barycentric coordinates are barycentric coordinates that are defined by rational functions of minimal degree. We study the rational map on P2 defined by Wachspress barycentric coordinates, the Wachspress map, and we describe polynomials that set-theoretically cut out the closure of the image, the Wachspress variety. The map has base points at the intersection points of non-adjacent edges. The Wachspress map embeds the polygon into projective space of dimension one less than the number of vertices. Adjacent edges are mapped to lines meeting at the image of the vertex common to both edges, and base points are blown-up into lines. The deformed image of the polygon is such that its non-adjacent edges no longer intersect but both meet the exceptional line over the blown-up corresponding base point. We find an ideal that cuts out the Wachspress variety set-theoretically. The ideal is generated by quadratics and cubics with simple expressions along with other polynomials of higher degree. The quadratic generators are scalar products of vectors of linear forms and the cubics are determinants of 3 x 3 matrices of linear forms. Finally, we conjecture that the higher degree generators are not needed, thus the ideal is generated in degrees two and three.

<p>In this thesis we try to construct a stability condition on the quintic threefold. We have not succeeded in proving the existence of such a stability condition. However we have constructed a stability condition on a quotient category of projective space that approximates the quintic. We conjecture the existence of a stability condition on the quintic threefold generated by spherical objects and explore some consequences.</p> / Dissertation

In this thesis, we first summarize the known results about lifting algebraic surfaces in characteristic p &gt / 0 to characteristic zero, and then we study lifting fibrations on these surfaces to characteristic zero. We prove that fibrations on ruled surfaces, the natural fibration on Enriques surfaces of classical type, the induced fibration on K3-surfaces covering these types of Enriques surfaces, and fibrations on certain hyperelliptic and quasi-hyperelliptic surfaces lift. We also obtain some fragmentary results concerning the smooth isotrivial fibrations and the fibrations on surfaces of Kodaira dimension 1.

Hypergeometric functions seem to be ubiquitous in mathematics. In this document, we present a couple of ways in which hypergeometric functions appear in arithmetic geometry. First, we show that the number of points over a finite field [mathematical symbol] on a certain family of hypersurfaces, [mathematical symbol] ([lamda]), is a linear combination of hypergeometric functions. We use results by Koblitz and Gross to find explicit relationships, which could be useful for computing Zeta functions in the future. We then study more geometric aspects of the same families. A construction of Dwork's gives a vector bundle of deRham cohomologies equipped with a connection. This connection gives rise to a differential equation which is known to be hypergeometric. We developed an algorithm which computes the parameters of the hypergeometric equations given the family of hypersurfaces. / text

The thesis considers two distinct strategies for algebraic computation with polynomials in high dimension. The first concerns ideals and varieties with symmetry, which often arise in applications from areas such as algebraic statistics and optimization. We explore the commutative algebra properties of such objects, and work towards classifying when symmetric ideals admit finite descriptions including equivariant Gröbner bases and generating sets. Several algorithms are given for computing such descriptions. Specific focus is given to the case of symmetric toric ideals. A second area of research is on problems in numerical algebraic geometry. Numerical algorithms such as homotopy continuation can efficiently compute the approximate solutions of systems of polynomials, but generally have trouble with multiplicity. We develop techniques to compute local information about the scheme structure of an ideal at approximate zeros. This is used to create a hybrid numeric-symbolic algorithm for computing a primary decomposition of the ideal.

Morrison has constructed a geometric relationship between K3 surfaces with large Picard number and abelian surfaces. In particular, this establishes that the period spaces of certain families of lattice polarized K3 surfaces (which are closely related to the moduli spaces of lattice polarized K3 surfaces) and lattice polarized abelian surfaces are identical. Therefore, we may study the moduli spaces of such K3 surfaces via the period spaces of abelian surfaces. In this thesis, we will answer the following question: from the moduli space of abelian surfaces with endomorphism structure (either a Shimura curve or a Hilbert modular surface), there is a natural map into the moduli space of abelian surfaces, and hence into the period space of abelian surfaces. What sort of relationship exists between the moduli spaces of abelian surfaces with endomorphism structure and the moduli space of lattice polarized K3 surfaces? We will show that in many cases, the endomorphism ring of an abelian surface is just a subring of the Clifford algebra associated to the N\'eron-Severi lattice of the abelian surface. Furthermore, we establish a precise relationship between the moduli spaces of rank 18 polarized K3 surfaces and Hilbert modular surfaces, and between the moduli spaces of rank 19 polarized K3 surfaces and Shimura curves. Finally, we will calculate the moduli space of E_8^2 + <4>-polarized K3 surfaces as a family of elliptic K3 surfaces in Weierstrass form and use this new family to find families of rank 18 and 19 polarized K3 surfaces which are related to abelian surfaces with real multiplication or quaternionic multipliction via the Shioda-Inose construction. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-08-12 14:38:04.131

In this thesis, we study the problem of lifting fibrations on surfaces in characteristic p, to characteristic zero. We restrict ourselves mainly to the case of natural fibrations on surfaces with Kodaira dimension -1 or 0. We determine whether such a fibration lifts to characteristic zero. Then, we try to find the smallest ring over which a lifting is possible. Finally,in some favourable cases, we compare the moduli of liftings of the fibration to the moduli of liftings of the surface under consideration.

Thesis (Ph. D.)--University of Hong Kong, 2008. / Includes bibliographical references (leaf 31-35) Also available in print.

Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Sept. 9, 2009). Vita. Includes bibliographical references.