Author ORCID Identifier

https://orcid.org/0000-0001-9548-3866

Date Available

4-18-2022

Year of Publication

2022

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Advisor

Dr. Benjamin Braun

Abstract

A long-standing conjecture in geometric combinatorics entails the interplay between three properties of lattice polytopes: reflexivity, the integer decomposition property (IDP), and the unimodality of Ehrhart h*-vectors. Motivated by this conjecture, this dissertation explores geometric properties of weighted projective space simplices, a class of lattice simplices related to weighted projective spaces.

In the first part of this dissertation, we are interested in which IDP reflexive lattice polytopes admit regular unimodular triangulations. Let Delta(1,q)denote the simplex corresponding to the associated weighted projective space whose weights are given by the vector (1,q). Focusing on the case where Delta(1,q) is IDP reflexive such that q has two distinct parts, we establish a characterization of the lattice points contained in Delta(1,q) and verify the existence of a regular unimodular triangulation of its lattice points by constructing a Grobner basis with particular properties.

In the second part of this dissertation, we explore how a necessary condition for IDP that relaxes the IDP characterization of Braun, Davis, and Solus yields a natural parameterization of an infinite class of reflexive simplices associated to weighted projective spaces. This parametrization allows us to check the IDP condition for reflexive simplices having high dimension and large volume, and to investigate h* unimodality for the resulting IDP reflexives in the case that Delta(1,q) is 3-supported.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2022.129

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