Tropical Data in Combinatorial Algebraic Geometry

• Applicant: Dr. Alheydis Geiger
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539846931

Mathematical Data becomes more important in particular within combinatorics. One aim of this project is to work with the data around the (tropical) Grassmannian and its subvarieties, as well as analogues like the moduli space of point configurations or the moduli space of del Pezzo surfaces of degree 3, and to further investigate this data with a focus on the connection of tropical geometry and combinatorics. The main feature in this investigation are matroids. The first step is collecting available data and computing new data around the tropical, chirotopal, positive or self-dual Grassmannian and making them available according to the FAIR data principles. In the following investigation a special focus will be given to self-dual positroids and the self-dual Grassmannian for rank 4 matroids. This latter object also plays a main role in the investigation of Mustafin degenerations of 3-dimensional projective space with respect to fixed points. It provides a smaller space of possible point configurations than the full moduli space. This analysis is also part of the proposal. For the moduli space of cubic del Pezzo surfaces, we want to focus particularly on the tropical positive aspects, i.e. which of the tropical cubic surfaces are positive. For the tropical Grassmannian Gr(2,n) positivity corresponds to planarity of a circularly labeled tree. We want to extend this identification to the tropical cubic surfaces, which can be identified by an arrangement of 27 trees. In order to understand a parameter or moduli space it is important to understand its compactification and boundary. This is the object of the investigation in the second to last proposed project of this proposal. A numerical method using tropical geometry seems to capture the stratification of the boundary of the moduli spaces of point configurations and of the moduli space of degrees 2 and 3 del Pezzo surfaces. The aim is to understand this correspondence better and to extend the technique to other compactifications to gain combinatorial insights. The last part is concerned with (self-dual) matroids from graph curves. Graph curves are combinatorial objects: curves that are up to projective transformation uniquely determined by their dual graphs, which are simple, 3-connected and trivalent. By means of generic hyperplane sections one can gain a self-dual matroid from such a graph curve. This matroid is not the graphical matroid of the associated graph nor its dual. This process can be explained by algebraic geometry. However, a full combinatorial understanding of the process, dependencies and causalities is missing. The aim is to rectify that.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies