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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