Understanding the Defining Ideal Through Cones

This dissertation investigates the properties of "cone polynomials" in algebraic geometry, focusing on when these polynomials can generate the defining ideal of a smooth projective variety. The primary area of study is the specific case where the variety X is a finite set of points in a 2-dimensional projective space (P²). The research explores the relationship between cone polynomials and the Castelnuovo-Mumford regularity of varieties. Cone polynomials offer a geometric method to derive equations that define a variety, and understanding their generating power helps in bounding the algebraic complexity of these varieties. The work builds upon previous research, particularly concerning generic points in P² and the ideal generated by cone polynomials. A key finding presented is the result for collinear points in P², which establishes that the ideal generated by cone polynomials matches the defining ideal of the points for degrees k greater than or equal to 2d-2, where d is the number of points. This result is extended to arbitrary finite sets of points in P² through deformation arguments, providing an upper bound for the saturation degree.

Curriculum

The dissertation is structured into several chapters covering background material, the introduction of cone polynomials, and the main results.

1. Background
   • Saturation
   • Macaulay's Theorem
   • Regularity
   • Flatness
2. Cone Polynomials
   • Definitions and Motivation
   • Theorems of Fu and Nie
   • Cutting out Varieties with Cone Polynomials
   • A Preliminary Bound for the Saturation Degree of the Cone Ideal
3. Main Results for Finite Sets in the Plane
   • Collinear Points
   • Implications for Non-collinear Points
4. Experimental Data and Further Directions
   • Cones over Points in the Plane
   • Cones over Points in Higher Projective Spaces
   • Cones over Curves
   • Further Directions