Tropical Singularities
Final Report Abstract
Tropical geometry can be viewed as a convex geometry shadow of algebraic geometry. Tropicalization is a process that associates a rational polyhedral complex satisfying certain properties to an (embedded) algebraic variety. Under tropicalization, many features are preserved, leading to an infusion of convex geometry tools into algebraic geometry. In order to make use of this new toolkit, many basic questions concerning the nature of tropicalizations have to be answered. This project dealt with such a basic question, namely the tropical analogue of singularities of algebraic varieties. Before the project started, tropical analogues of nodes for hypersurfaces had been studied, revealing interesting combinatorial and metric features. The project focused on three main directions: (a) more singular points resp. enumerative results concerning singular tropical hypersurfaces, (b) the positive part of the discriminant, resp. real tropical singularities, and (c) other singularity types. In part (a), the famous lattice path algorithm to count tropical curves inspired several research directions. In a joint project with Erwan Brugallé, I investigated enumerative formulas for numbers of curves on Hirzebruch surfaces satisfying point conditions. These formulas are obtained by a tropical degeneration technique involving counts of tropical curves with a singularity in the toric boundary. The necessary correspondence theorems are proved to make the formular valid for numbers of algebraic curves. With my coauthor Marίa Angélica Cueto, I developed a technique that is suitable to embed tropical curves in other models of the tropical plane, getting closer to the Berkovich skeleton. The curve in question must have a tropical singularity, and the algorithm can be viewed as a way to resolve the singularity in the initial degeneration. Relying on our classification of singular tropical surfaces, my coauthors Thomas Markwig and Eugenii Shustin and I constructed a three-dimensional version of the lattice path count. We also deliver the relevant correspondence theorem. Our techniques are easily adapted to real algebraic geometry: with a slight modification of the tropical multiplicities, we can use exactly the same tropical counting technique to count real singular surfaces satisfying incidence conditions. In part (b) of the project, my PhD student Christian Jürgens obtained a classification of real tropical plane curves and real tropical surfaces in threespace with a singularity in a real point, combining methods involving signed Bergman fans and tropical classification methods. The signed Bergman fans are generalizations of positive Bergman fans studied before. In part (c), he obtained partial results involving plane tropical curves with a cusp resp. with an m-fold point.
Publications
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Deformation of tropical Hirzebruch surfaces and enumerative geometry. J. Algebraic Geom. 25 (2016), no. 4, 633–702
Erwan Brugallé , Hannah Markwig
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How to repair tropicalizations of plane curves using modifications. Exp. Math. 25 (2016), no. 2, 130–164
Marίa Angélica Cueto, Hannah Markwig
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Enumeration of Complex and Real Surfaces via Tropical Geometry. Adv. Geom. 18 (2018), no. 1, 69–100
Hannah Markwig, Thomas Markwig, Eugenii Shustin
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Real tropical singularities and Bergman fans
Christian Jürgens
