Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Bernd Ulrich

Committee Chair

Bernd Ulrich

Committee Member 1

Giulio Caviglia

Committee Member 2

Edray H. Goins

Committee Member 3

William J. Heinzer


Consider the rational map Ψ : [Special characters omitted.] where the fi's are homogeneous forms of the same degree in the homogeneous coordinate ring R = k[ x1,…,xd] of [Special characters omitted.]. Assume that I = (f 1,…,fm) is a height 2 perfect ideal in the polynomial ring R. In this context, the coordinate ring of the graph of Ψ is the Rees algebra of I and the co-ordinate ring of the image of Ψ is the special fiber ring. We study two settings. The first setting is when I is almost linearly presented. Here we study the ideal defining the graph and the image of Ψ. Whenever possible, we also study invariants such as the Castelnuovo-Mumford regularity and the relation type of the graph of Ψ. In the second setting we impose no constraints on the column degrees of the presentation matrix of I, but the number of generators of I is restricted to d + 1 (two more than the dimension of the source of Ψ). For this configuration, we study the image of Ψ.

We also introduce a new method, namely "iterated'' Jacobian duals, to study the graph of Ψ. This is a generalization of the usual Jacobian duals which are often used to describe the graph of Ψ.

Included in

Mathematics Commons