#### Date of Award

8-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### 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

#### Abstract

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*[* x*1,…,*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.

#### Recommended Citation

Mukundan, Vivek, "Rees algebras and iterated Jacobian duals" (2016). *Open Access Dissertations*. 818.

https://docs.lib.purdue.edu/open_access_dissertations/818