
## 3. Geometric aspects

1. #### Problem 3.02.

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.
• Do flat morphisms of toric varieties tropicalize to tropically flat morphisms?
• Flatness should be easy to describe for tropical abelian varieties, in that the naive definition should work.
• #### Problem 3.04.

What is a smooth subscheme of a tropical scheme? What is a smooth point?
• #### Problem 3.06.

What is the multiplicity of a point on a tropical scheme?
• #### Problem 3.08.

What are differential forms on tropical schemes?
• #### Problem 3.1.

Define the tangent space of a tropical scheme.
• #### Problem 3.12.

Does the balancing condition hold for arbitrary (i.e. non-realizable) tropical schemes?
• #### Problem 3.14.

Is there projective duality for tropical varieties?
• #### Problem 3.16.

Are there Chow forms in tropical geometry?
In the realizable case, Maclagan’s student Tripoli has tropicalized Chow forms.
• #### Problem 3.18.

How can the tropical scheme structure change under tropical modifications?
This is a special case of summing tropical ideals.
• #### Problem 3.2.

Can we see faithful tropicalizations on the level of schemes?
• #### Problem 3.22.

What are Deligne–Mumford stacks over tropical schemes?
•     A definition of a category of tropical schemes should have three components:
1. A category of affine schemes;
2. Charts (Zariski localizations);
3. Covering conditions.
We have good candidates for the first two. Here is a proposal for the third.

Let $\Delta$ be a compact rational polytope. Let $\text{CPA}(\Delta)$ denote convex piecewise-affine functions on $\Delta$. There are two reasonable definitions for when a finite collection of sub-polytopes covers $\Delta$:
1. (a) They cover $\Delta$ and their interiors cover the interior of $\Delta$.
2. (b) We have descent for $\text{CPA}(\Delta)$-modules.
It is true that (b) implies (a).

#### Problem 3.24.

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.

Develop a way to tropicalize subschemes of toric vector bundles.
This could give a way to coordinatize coherent sheaves on a subscheme of a toric variety, and therefore lead to a definition of coherent sheaves on tropical schemes (at least in the realizable case).
• #### Problem 3.3.

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 Hilbert-Chow morphism.
• #### Problem 3.4.

Find a way to recover/detect properties (reducedness, smoothness, irreducibility, number of components) of a scheme from its tropicalization.
• #### Problem 3.42.

What is the étale topology for tropical schemes? Étale maps?
• #### 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 Joó-Mincheva’s tropical Krull dimension.

Cite this as: AimPL: Foundations of tropical schemes, available at http://aimpl.org/tropschemes.