Graduate Studies

 

First Advisor

Alexandra Seceleanu

Degree Name

Doctor of Philosophy (Ph.D.)

Committee Members

Berthe Choueiry, Jack Jeffries, Mark Walker

Department

Mathematics

Date of this Version

8-2025

Document Type

Dissertation

Citation

A dissertation presented to the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Alexandra Seceleanu

Lincoln, Nebraska, August 2025

Comments

Copyright 2025, Shahriyar Roshan Zamir. Used by permission

Abstract

Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree $d$ singular at a given set of points.

After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this thesis, we use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space.

A main contribution of this work is the careful adaption of several classical algebro-geometric techniques to the setting of weighted projective geometry. The introduction is dedicated to non-technical remarks about interpolation including its history. Chapter 2 summarizes the results. Chapter 3 includes the set up and most of the necessary background. Chapters 4–8 contain the main results and Chapter 9 contains some open problems.

Advisor: Alexandra Seceleanu

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