<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.

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2005. / Includes bibliographical references.

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