Combinatorial and Tropical Degenerations of del Pezzo Surfaces and Their Moduli

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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. This research project will provide a new combinatorial perspective on a classical family of moduli spaces, namely del Pezzo surfaces. 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. New findings will contribute to the rapidly growing open source mathematical software Sage. 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. The objective of this research is to develop effective combinatorial methods to study moduli space of del Pezzo surfaces and related geometric objects from the perspective of degenerations, tropical and nonarchimedean geometry. The project has three components. First, to establish effective methods for constructing faithful tropicalization and characterizing the combinatorics of del Pezzo surfaces and their moduli in low degrees to harness geometric and topological invariants via polyhedral combinatorics. Second, to interpret well-known classical enumerative geometry results on del Pezzo surfaces of low degrees in the tropical setting. Finally, the PI will apply polyhedral techniques to establish new obstructions to lifting curves from the tropical to the algebraic world, addressing "superabundance" of tropical curves, and explore questions involving unirationality and transition maps between various known coordinate systems on del Pezzo surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.