Problems in Computational Algebraic Geometry: Lefschetz Properties and Toric Varieties

Abstract

The first chapter gives the mathematical background needed for the subsequent chapters. The second chapter consists of the publication [16], a joint work with Hal Schenck, in which the we examine the influence of geometry on the weak Lefschetz property. We show that with a certain configuration of points in projective space, their Artnian reduction does have the weak Lefschetz property. We then generalize this result to a Boij-S¨oderberg theoretic condition on Betti tables of Artinian algebras.

In the third chapter, the focus shifts to toric varieties and results on the Castelnuovo- Mumford regularity of toric surfaces. Inspired by a result of L’vovsky in 1996, we show that the combinatorics of a projective toric surface can provide a combinatorial bound on the Castelnuovo-Mumford regularity. An overview is given on the tools used to approach this problem, and the chapter closes with some open questions.