A Zariski-Nagata Theorem for Smooth Toric Surfaces

Authors

• Jordan Vincent Barrett, University of Nebraska-LincolnFollow

First Advisor

Jack Jeffries

Degree Name

Doctor of Philosophy (Ph.D.)

Committee Members

Alexandra Seceleanu, Eloísa Grifo, Mona Bavarian

Department

Mathematics

Date of this Version

5-5-2025

Document Type

Dissertation

Citation

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 Jack Jeffries

Lincoln, Nebraska, May 2025

Comments

Copyright 2025, Jordan Vincent Barrett. Used by permission

Abstract

The Affine Zariski-Nagata theorem is a classical result in commutative algebra that gives an expression for the n^th symbolic power of a radical ideal I in a polynomial ring over a field in terms of the n^th ordinary powers of the maximal ideals in [affine variety]_max(I). In this thesis we discuss a well-known projective analog of Zariski-Nagata and provide the necessary background on toric varieties to present a generalization of this result to toric surfaces. We conclude with a brief discussion about work toward characterizing which abstract toric varieties have smooth point fibers with the aim of understanding the symbolic powers of higher dimensional toric varieties.

Advisor: Jack Jeffries

Recommended Citation

Barrett, Jordan Vincent, "A Zariski-Nagata Theorem for Smooth Toric Surfaces" (2025). Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–. 294.
https://digitalcommons.unl.edu/dissunl/294