# Equations of Rees Algebras and Singularities of Rational Plane Curves

This dissertation examines two related topics, defining equations of Rees algebras in two variables and the geometry of rational plane curves. First, we consider the ideal K of defining equations of the Rees algebra of an ideal I in the polynomial ring k[x_1,x_2] generated by forms of the same degree. We give a description of the part of K in degree greater than or equal to the second largest degree of a generator of the syzygy module of I. This description lets us compute the minimal generators of K in these degrees. When I is generated by three forms, we also produce a generating set for each graded slice K_i in this range.

The second topic we study is the geometry of birationally parametrized plane curves C, particularly their singularities. This is connected to the first part because the parametric equations generate an ideal I in k[x_1,x_2], and the Rees algebra of I is the coordinate ring of the graph of the parametrization. We show that the Fitting ideals of a slice of the symmetric algebra of I detect the singular points on or infinitely near to C as well as their multiplicities. Using this, we stratify the space of Hilbert-Burch matrices of parametrizations into locally closed subsets representing those curves with a given number of points of each multiplicity. Finally, we connect one of these Fitting ideals to a slice of the ideal K. In particular, we give a complete list of the possible bidegrees of a minimal generating set of K when the degree of C is 7, as well as the correspondence between these bidegrees and the multiplicities of the points of C, which was previously known only for degree <= 6.

## History

## Date Modified

2017-06-02## Defense Date

2016-03-31## Research Director(s)

Claudia Polini## Committee Members

Claudiu Raicu Juan Migliore Sonja Mapes## Degree

- Doctor of Philosophy

## Degree Level

- Doctoral Dissertation

## Program Name

- Mathematics