Combinatorial and Tropical Degenerations of Classical Moduli Spaces

Ohio State University, The, Columbus OH

Investigators

Abstract

Tropical geometry is a young and rapidly growing area in mathematics, rooted in algebraic geometry, complex analysis, commutative algebra, and combinatorics, with applications in computer science, biology, and statistical physics, in addition to other areas of mathematics. The recent decade has seen tremendous development in the subject that both established the field as an area in its own right and unveiled its deep connections to numerous branches of pure and applied mathematics. Moduli spaces are spaces of parameters that classify algebro-geometric mathematical objects, such as varieties, maps between varieties, and vector bundles. This research project aims at providing a new combinatorial perspective on classical moduli spaces and their interplay with nonarchimedean analytic geometry. Computational challenges in this direction are its driving force. The primary goal is to deepen understanding of these objects and develop new techniques to analyze such abstract spaces through concrete computations in tropical geometry. Several topics investigated in this project witness the interdisciplinary nature of the subject and are suitable for research in collaboration with graduate students. Tropical geometry provides a framework for solving algebro-geometric problems using concrete combinatorial tools: algebraic varieties are replaced by weighted, balanced polyhedral complexes. These objects preserve just enough data about the original varieties to remain meaningful, while discarding much of their complexity. Their combinatorics depends strongly on the embeddings of our varieties. By contrast, the Berkovich space or the space of valuations is independent of all such choices. By choosing appropriate coordinates, the geometry and topology of these valuations are reflected in rich combinatorial properties on the polyhedral side, including connectedness, shellability, etc. Thus, the intricacies of the original space turn into exciting combinatorial problems. Tropicalizations resulting from these nice embeddings are said to be faithful. The project aims at developing certificates for local faithfulness by means of initial (Groebner) degenerations of the variety. The notion of faithfulness near that point then turns into desirable properties of this degeneration. Not surprisingly, validating these properties requires a deep understanding of the combinatorics and geometry of the initial degenerations. Thus, the need to find effective methods for constructing or detecting faithfulness is at the core of tropical geometry. The project will develop such program for several important classes of examples: Grassmannians, moduli of curves, del Pezzo surfaces, Fano schemes and cluster varieties.