Topics in Tropical and Analytic Geometry Open Access

Abstract

In this thesis, the author proves theorems on the existence and mapping properties of tropical stacks that arise from the theory of toric artin stacks. The author also provides a generalization, using the same ideas from toric artin stacks, of recent work involving analytic stacks and their tropicalizations. The author also proves a result comparing the tropicalization of the jet bundle to the jet bundle of the tropicalization.

Table of Contents

1 Introduction 1 2 Background 4 2.1 Classical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 General construction . . . . . . . . . . . . . . . . . . . 4 2.1.2 Polyhedral connections . . . . . . . . . . . . . . . . . . 7 2.1.3 Analytic picture . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Scheme Theoretic . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 F1 and semiring algebra . . . . . . . . . . . . . . . . . 11 2.2.2 Tropical schemes . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Examples of the differences between using rings and semirings . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Jet Bundles of Tropical Hypersurfaces 15 3.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Tropical Connections . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 All Things Stacks 19 4.1 All Things Stacks . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1.1 Main definitions . . . . . . . . . . . . . . . . . . . . . . 19 4.2 All Things Toric . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.1 Toric Geometry . . . . . . . . . . . . . . . . . . . . . . 21 4.2.2 GS toric stacks . . . . . . . . . . . . . . . . . . . . . . 23 4.2.3 Tropical Toric Stacks . . . . . . . . . . . . . . . . . . . 27 4.3 All Things Analytic . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3.1 Analytic Geometry . . . . . . . . . . . . . . . . . . . . 30 4.3.2 Ulrisch’s Work . . . . . . . . . . . . . . . . . . . . . . 35 4.3.3 Extending Ulrisch’s work . . . . . . . . . . . . . . . . . 36 Bibliography 38

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	Algebraic Geometry
Committee Chair / Thesis Advisor	Zureick-Brown, David, Emory University
Committee Members	Powers, Victoria, Emory University Duncan, John, Emory University

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