## Mathematics, Department of

## First Advisor

Brian Harbourne

## Date of this Version

Spring 5-5-2023

## Document Type

Article

## Citation

Kettinger, Jake, "On the Superabundance of Singular Varieties in Positive Characteristic" (2023). *ETD collection for University of Nebraska - Lincoln*. AAI30489213. https://digitalcommons.unl.edu/dissertations/AAI30489213

## Abstract

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not known what other examples there can be. Furthermore, before the work in this thesis, only a few examples of geproci nontrivial non-grid non-half-grids were known and there was no known way to generate more. Here, we use geometry in the positive characteristic setting to give new methods of producing geproci half-grids and non-half-grids. We also pick up work that had been done in 2017 by Solomon Akesseh, who had proven that there are no unexpected cubics in characteristic 3 with distinct points and gave examples involving infinitely near points based on quasi-elliptic fibrations in characteristic 2. Each quasi-elliptic fibration has a Dynkin diagram. Here, in contrast, for each possible Dynkin diagram for a quasi-elliptic fibration in characteristic 3, we give an example of the fibration but show it does not give rise to an unexpected cubic. [Equations Omitted]

Adviser: Brian Harbourne

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Brian Harbourne. Lincoln, Nebraska: May, 2023

Copyright © 2023 Jake Kettinger