Unexpectedness Stratified by Codimension

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 Vallés and how they were used in the 2018 paper of Cook, 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. Advisor: Brian Harbourne