On the Fattening of Lines in $\mathbf{P}^3$

TL;DR

This paper investigates the classification of arithmetically Cohen-Macaulay codimension 2 subschemes in P^3 using the invariant alpha, extending prior classifications from P^2 to three-dimensional projective space.

Contribution

It introduces a classification approach for these subschemes in P^3 based on the invariant alpha, building on previous work in P^2.

Findings

Classification of arithmetically Cohen-Macaulay codimension 2 subschemes in P^3

Relation of the invariant alpha to Hilbert functions and gamma

Extension of Bocci and Chiantini's methods to P^3

Abstract

We follow the lead of Bocci and Chiantini and show how differences in the invariant alpha can be used to classify certain classes of subschemes of P^3. Specifically, we will seek to classify arithmetically Cohen-Macaulay codimension 2 subschemes of P^3 in the manner Bocci and Chiantini classified points in P^2. The first section will seek to motivate our consideration of the invariant alpha by relating it to the Hilbert function and gamma, following the work of Bocci and Chiantini, and Dumnicki, et. al. The second section will contain our results classifying arithmetically Cohen-Macaulay codimension 2 subschemes of P^3. This work is adapted from the author's Ph.D. dissertation.