10.13023/ETD.2020.308
Trok, William
William
Trok
University of Kentucky
Geometry of Linear Subspace Arrangements with Connections to Matroid Theory
University of Kentucky Libraries
2020
Dissertation
FOS: Mathematics
This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond's about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points.
We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface'' passing through a fixed set of points Z. We show in certain cases these can be characterized via combinatorial data and geometric data from the Hyperplane Arrangement dual to Z. This generalizes earlier results on unexpected curves in the plane due to Faenzi, Valles, Cook, Harbourne, Migliore and Nagel.
© 2020 William Trok