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## 1. Applications and long-term goals

1. ### Clemens conjecture

#### Conjecture 1.05.

A smooth quintic threefold has only finitely many rational curves of each degree.
As a warm up, describe the family of lines on a tropical cubic surface missing from Vigeland’s paper.
• ### Nagata’s conjecture on curves

#### Conjecture 1.1.

Any curve passing through $n > 9$ general points in $\mathbb{P}^2$ with multiplicities $m_1, \ldots, m_n$ must have degree greater than $\frac{1}{\sqrt{n}} \sum_{i=1}^n m_i.$
Approach: Show that the conjecture holds tropically, and that any counterexample would tropicalize to a tropical counterexample.
• ### Hartshorne’s conjecture on vector bundles

#### Conjecture 1.15.

Every rank $2$ vector bundle on $\mathbb{P}^n$, $n \geq 7$, splits as a sum of line bundles.
Approach: Develop a theory of tropical vector bundles.
• ### Moduli space of metric graphs

The rational cohomology of the moduli space of metric graphs is zero above its “virtual cohomological dimension,” and is also zero in a stable range. Together, this means this moduli space is a formal space in the sense of rational homotopy theory.

#### Problem 1.2.

Does tropical geometry give us a reason why the moduli space of metric graphs should be a formal space?

• Motivation: Complex Kähler manifolds are formal.
• The nonarchimedean Poincaré-Lelong formula may be useful here. Is there a way to apply it?
• ### Realization problem for tropical curves

#### Problem 1.25.

Can tropical scheme structure distinguish between realizable and unrealizable tropical curves (in $\mathbb{R}^3$, e.g.).
This has applications to classical questions, e.g., how many quintics in $\mathbb{P}^2$ are tangent to 20 specified lines?
• ### Hilbert schemes of points

#### Problem 1.3.

Construct and compare $\text{Trop}(\text{Hilb}^n(X))$ and $\text{Hilb}^n(\text{Trop}(X))$.
• #### Problem 1.35.

Given 2 monomial ideals, when is there a connected smooth curve $(\mathbb{P}^1 \hookrightarrow \text{Hilb}^n(\mathbb{C}^2))$ joining them?
• #### Conjecture 1.4.

$\text{Hilb}^n(\mathbb{C}^3)$ is generically reduced.
Two possible approaches:
1. Compute the scheme-theoretic tropicalization of $\text{Hilb}^n(\mathbb{C}^3)$.
2. Show that the Behrend function is constant. The Behrend function can be calculated by motivic integration, which can be done tropically.
• #### Problem 1.45.

More generally, when are Hilbert schemes of points on threefolds reducible?
• ### Brill-Noether theory

#### Problem 1.5.

Develop a tropical version of limit linear series.
As a test case, try to recover the recent work of Chan-Pfleuger.
• ### Tropical Euler characteristic

#### Problem 1.55.

For a tropical scheme $Y$, define $\chi(\mathcal{O}_Y)$.
Candidate: $H_Y(0)$, where $H_Y$ is the Hilbert polynomial.
• ### Abelian varieties

#### Problem 1.6.

What is a tropical abelian variety? How do we tropicalize subschemes of abelian varieties?
• #### Problem 1.65.

Extend the results of Giansiracusa-Giansiracusa’s first paper to maximally degenerate abelian varieties.
• ### Combinatorics

#### Problem 1.7.

Which questions in combinatorics are amenable to tropical scheme theory?
• #### Problem 1.75.

Use tropical scheme theory to approach MacPherson’s conjecture that the “space of oriented matroids” is homotopy equivalent to the real oriented Grassmannian.
A proof was claimed, then shown wrong. A claimed counterexample [Gaku Liu] surfaced recently (status unknown). Even if the counterexample is valid, there are generalizations of this conjecture which may be amenable to tropical scheme theory.
• ### Cohomology and intersection theory

#### Problem 1.8.

Develop a theory of cohomology for tropical schemes.
• #### Problem 1.85.

Does tropical cohomology satisfy Hard Lefschetz? Can we use it to approach problems in combinatorial algebraic geometry?

• Can the recent proof of the Rota-Heron-Welsh conjecture (by Adiprasito-Huh-Katz) and its machinery be understood in the context of tropical schemes?
• Use tropical schemes to approach the top-heavy conjecture of Dowling-Wilson in the non-realizable case (recently proved for realizable matroids by Huh-Wang).
• #### Problem 1.9.

Develop scheme-theoretic tropical intersection theory.

• Intersection theory for tropical varieties has been quite successful.
• Is there a tropical version of Serre’s Tor-formula for intersection multiplicities?
• #### Problem 1.95.

Prove a tropical Riemann-Roch theorem by cohomological methods.
Problem: What is the scheme-theoretic tropicalization of a semistable model?

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