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3. Geometric aspects


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      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.


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          Problem 3.04.

          What is a smooth subscheme of a tropical scheme? What is a smooth point?


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              Problem 3.06.

              What is the multiplicity of a point on a tropical scheme?


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                  Problem 3.08.

                  What are differential forms on tropical schemes?


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                      Problem 3.1.

                      Define the tangent space of a tropical scheme.


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                          Problem 3.12.

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


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                              Problem 3.14.

                              Is there projective duality for tropical varieties?


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                                  Problem 3.16.

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


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                                      Problem 3.18.

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


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                                          Problem 3.2.

                                          Can we see faithful tropicalizations on the level of schemes?


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                                              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).

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                                                  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


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                                                      Problem 3.26.

                                                      Endow the skeleton of a Berkovich curve with a tropical scheme structure.


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                                                          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).


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                                                              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


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                                                                  Problem 3.32.

                                                                  Define the embedded tangent space to a surface in $\mathbb{TP}^3$.


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                                                                      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?


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                                                                          Problem 3.36.

                                                                          Find a modular interpretation of the tropical Dressian and tropical Grassmannian.


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                                                                              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.


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                                                                                  Problem 3.4.

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


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                                                                                      Problem 3.42.

                                                                                      What is the étale topology for tropical schemes? Étale maps?


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                                                                                          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?


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                                                                                              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.