Michael Joswig |
Hannah Markwig |
Thomas Markwig |
We will study moduli spaces of (abstract and embedded) tropical curves and surfaces. The computation of such moduli spaces is highly non-trivial and poses challenges which are interesting to master.
In its beginnings, embedded tropical varieties (arising as tropicalisations of embedded algebraic varieties) were at the centre of attention of tropical geometry. For the application of tropical methods in algebraic geometry, the dependence of tropicalisation on the embedding in general is a difficulty. For curves, the interplay of Berkovich analytifications, semi-stable reductions over discrete valuation rings and tropical geometry has led to an understanding of abstract tropical geometry relying on an abstract tropicalisation independent of a chosen embedding. The tropical geometry of curves nowadays is a beautiful and fruitful research area linking different areas like algebraic geometry, combinatorics, number theory, convex geometry, and optimisation. Moduli spaces of abstract tropical curves have been studied and feature strong ties with their algebro-geometric counterparts. Only recently, in a preprint of the first PI with Brodsky, Morrison and Sturmfels [1], the study of moduli spaces of (planar) embedded tropical curves was initiated, in comparison with spaces of abstract tropical curves. This study largely follows a computational approach, the challenge to compute large examples of such moduli spaces and maps between them is attacked and overcome for curves of genus up to four (with partial results for genus five). It is observed that moduli spaces of planar tropical curves do not behave exactly as one would expect from the perspective of algebraic geometry. In this sense, the moduli spaces of planar tropical curves considered in [1] are “incomplete”, we are missing certain tropical curves which should be added to restore strong ties with algebraic geometry.
In our project, we suggest two methods to enlarge the existing moduli spaces of planar tropical curves: modifications and compactifications. We intend to apply these techniques for moduli spaces of curves of genus two (in connection with Igusa invariants as appearing in number theory) and three. We envision an algorithmic treatment of this problem enabling us to compute examples as necessary for our investigation. Furthermore, we plan to attack moduli spaces of tropical surfaces, both with the tools already existing for curves as with the tools to be sharpened for curves within this project. While the computational challenges will be even bigger, the TRR 195 provides the right framework to attack these issues. We will employ Singular, polymake and the interface between them.
Polymake 3.2, 2018