Birational Geometry of Moduli Spaces and Bridgeland Stability

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Important systems in many areas of science and engineering, ranging from evolutionary biology to physics and from cryptography to computer science, can be described by systems of polynomial equations. Algebraic geometry studies solutions of such polynomial systems. Often, the solutions of these systems vary in a continuous manner depending on parameters in the defining equations of these systems, leading to parametrized spaces of solutions. A difficult-to-study system may lie in the same parameter space as a system that is easy to solve, allowing one to deduce geometric constraints on the difficult system. This project studies the geometry of some parameter spaces that are ubiquitous in mathematics and physics, so-called moduli spaces of vector bundles and moduli spaces of curves. The research computes geometric invariants of these spaces by relating them to simpler spaces. A recent breakthrough called Bridgeland stability has allowed computation of many of the invariants that were previously inaccessible. This research project aims to extend these results. In addition, the project contributes to training the next generation of researchers in mathematics through the involvement of graduate students and postdoctoral associates in the research. Bridgeland stability conditions provide a new tool for studying the geometry of moduli spaces of sheaves. This project aims to combine Bridgeland stability conditions with recent advances in birational geometry to study longstanding questions about the geometry of moduli spaces of sheaves on surfaces and moduli spaces of curves. These moduli spaces play a central role in many branches of mathematics. Consequently, it is crucial to understand their cohomology and birational geometry. The project involves three parts. First, building on work with collaborators, the PI will study the birational geometry of the moduli spaces of sheaves on surfaces. He will compute their birational invariants such as the cones of ample, movable, and effective divisors, concentrating on Hirzebruch surfaces, del Pezzo surfaces, and certain surfaces of general type. Second, the PI will study resolutions of ideal sheaves of points and curves in projective space in terms of other exceptional collections. In the case of the plane, the geometry is significantly simplified if one resolves sheaves in terms of an appropriately chosen exceptional collection; this work will apply the same ideas to sheaves on higher-dimensional projective spaces. Third, the PI will continue collaborative projects on the effective cones of higher codimension cycles on moduli spaces of curves, the Kawamata-Morrison Cone Conjecture, and extremality properties of cycles in homogeneous spaces.