Doctor of Philosophy, The Ohio State University, 2010, Mathematics

This thesis studies the geometric and deformational behavior of linear series under degenerations with the aim of attacking the maximal rank conjecture. There are three parts. The first part gives an explicit construction of the classical tangent-obstruction theory for deformations of the pair (X,L) to the case when X is local complete intersection scheme and L a line bundle on X. In the second part, we propose a new method, using deformation theory, to study the maximal rank conjecture. We prove that the maximal rank conjecture holds for the first unknown case: line bundles of extremal degree. Problems related to the maximal rank conjecture have become potentially accessible to this new method. In the third part, a canonical semi-stable degeneration of the d-th symmetric product C(d) as the curve C becomes singular is constructed.

Committee: Herb Clemens (Advisor); James Cogdell (Committee Member); Roy Joshua (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2010, Mathematics

The notion of gradient ideals in a power series algebra over a noetherian local ring is defined with basic properties studied. A natural generalization, “multi-gradient ideals”, and the algebra of potential functions associated to an arbitrary ideal are introduced. Classification of multi-gradient ideals is given in the cases of principal ideals and monomial ideals by studying their algebras of potential functions. Abstract examples of non-multi-gradient ideals and geometric examples of multi-gradient ideals are constructed.

Committee: Herbert Clemens (Advisor); James Cogdell (Committee Member); Hsian-Hua Tseng (Committee Member) Subjects: Mathematics