The Cayley-Bacharach Condition 公开
Nair, Rohan (Summer 2025)
Abstract
A set of points S in n-dimensional complex projective space is said to satisfy the Cayley-Bacharach condition with respect to degree r hypersurfaces, or is CB(r), if any degree r hypersurface containing all but one point of S contains the final point. In recent literature, the condition has played an important role in computing a birational invariant called the degree of irrationality. However, the condition itself has not been studied extensively, and surprisingly little is known about the behavior of CB(r) sets. We will discuss a new approach to studying CB(r) sets, using combinatorial methods from matroid theory, and present some results to demonstrate how matroid theory can help us understand this condition in greater depth.
Table of Contents
1. Introduction
1.1 This Dissertation: Context and Motivation
1.2 Outline
2. The Cayley-Bacharach Condition
2.1 Some Historical Background
2.2 Definitions
2.3 Cohomological Interpretation
2.4 Examples
2.5 Measures of Irrationality
2.6 An Overview of Known Results
3. Some Relevant Matroid Theory
3.1 Matroids as Rank Functions
3.2 Independent and Dependent Sets
3.3 Bases, Circuits, and Connectivity
3.4 Rank Hyperplanes
3.5 Deletion and Restriction
4. Hilbert Function Matroids
4.1 A Matroid on Projective Points
4.2 Matroids and CB(r) Sets
4.3 Circuits and CB(r) Sets
4.4 Connected Hilbert Function Matroids
4.5 Future Research
About this Dissertation
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