3. Tropical scheme theory

Problem 3.02.
[Ulirsch] What is a morphism of tropical schemes? What is the right definition of flat, proper, smooth, étale? Under the naive definition of flatness (in terms of exactness), every tropical scheme fails to be flat over $\mathbb{T}$.
 J. Giansiracusa and Lorscheid have a notion of flat family of tropical linear spaces which preserves the Hilbert polynomial. Try to generalize this.
 [Ulirsch] Do flat morphisms of toric varieties tropicalize to tropically flat morphisms?
 [MacPherson] Flatness should be easy to describe for tropical abelian varieties, in that the naive definition should work.

Problem 3.04.
[Maclagan] What is a smooth subscheme of a tropical scheme? What is a smooth point? 
Problem 3.12.
[Lorscheid] Does the balancing condition hold for arbitrary (i.e. nonrealizable) tropical schemes? 
Problem 3.16.
[Maclagan] Are there Chow forms in tropical geometry? 
Problem 3.18.
[N. Giansiracusa] How can the tropical scheme structure change under tropical modifications? 
A definition of a category of tropical schemes should have three components:
 A category of affine schemes;
 Charts (Zariski localizations);
 Covering conditions.
Let $\Delta$ be a compact rational polytope. Let $\text{CPA}(\Delta)$ denote convex piecewiseaffine functions on $\Delta$. There are two reasonable definitions for when a finite collection of subpolytopes covers $\Delta$: (a) They cover $\Delta$ and their interiors cover the interior of $\Delta$.
 (b) We have descent for $\text{CPA}(\Delta)$modules.
Problem 3.24.
[MacPherson] Prove that (a) implies (b). If this fails to hold in general, does it hold for some subclass of $\text{CPA}(\Delta)$modules? 
Tropicalization in other settings
Problem 3.26.
Endow the skeleton of a Berkovich curve with a tropical scheme structure. 
Problem 3.28.
[Ulirsch] Develop a way to tropicalize subschemes of toric vector bundles. 
Problem 3.3.
[Ranganathan] Describe the relationship between tropicalization and perfect obstruction theories, using the tropicalization of the moduli space of stable maps as a guiding example. 
End of workshop problems
Problem 3.32.
Define the embedded tangent space to a surface in $\mathbb{TP}^3$. 
Problem 3.34.
Given an $n$marked genus $g$ tropical curve $\Gamma$, what is the smallest dimension of a toric variety $X$ such that a curve $C \hookrightarrow X$ gives a faithful tropicalization of $\Gamma$. Is there a universal faithful tropicalization? 
Problem 3.36.
Find a modular interpretation of the tropical Dressian and tropical Grassmannian. 
Problem 3.38.
Compute more examples of tropical Hilbert schemes (e.g. $\text{Hilb}^3(\mathbb{A}^1), \text{Hilb}^2(\mathbb{A}^2)$, …). Describe the HilbertChow morphism. 
Problem 3.4.
Find a way to recover/detect properties (reducedness, smoothness, irreducibility, number of components) of a scheme from its tropicalization. 
Problem 3.44.
Develop tropical scheme theory over the tropical hyperfield. Is this a better setting for thinking about prime ideals? Does the hyperfield topology help with localization? 
Problem 3.46.
Give a geometric interpretation of JoóMincheva’s tropical Krull dimension.
Cite this as: AimPL: Foundations of tropical schemes, available at http://aimpl.org/tropschemes.