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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.