Graduate Studies

 

First Advisor

Brian Harbourne

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

Date of this Version

12-7-2023

Document Type

Dissertation

Comments

Copyright 2023, Frank Zimmiti

Abstract

A recent series of papers, starting with the paper of Cook, Harbourne, Migliore and Nagel on the projective plane in 2018, studies a notion of unexpectedness for finite sets Z of points in N-dimensional projective space. Say the complete linear system L of forms of degree d vanishing on Z has dimension t yet for any general point P the linear system of forms vanishing on Z with multiplicity m at P is nonempty. If the dimension of L is more than the expected dimension of t−r, where r is N+m−1 choose N, we say Z has unexpected hypersurfaces of degree d and multiplicity m. We extend the definition of unexpectedness to include the possibility that the unexpectedness occurs only for P on a subvariety of positive codimension. We begin our study of this stratified unexpectedness by analyzing sets of points in the plane. We are able to give a characterization for unexpectedness for sets of points which lie on a degenerate conic. We then further analyze the techniques of Faenzi and Vall´es and how they were used in the 2018 paper of Cook II, Harbourne, Migliore, and Nagel. In the fourth chapter, we study lower bounds on the dimension of the space L. Lastly, we study the connection between Lefschetz properties and unexpectedness

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