Geometry of Moduli Spaces and Applications

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This proposal aims to study the geometry of moduli spaces, such as their dimensions, irreducible components, birational types, and geometric invariants. The principal investigator plans to carry out the study in several directions. Firstly, he would like to study Teichmueller curves, which are rigid geodesics in the moduli space of curves, and hence can provide crucial information for the geometry of the moduli space. Secondly, he wants to explore moduli spaces parameterizing curves in an ambient space, focusing on the comparison between the moduli space of stable maps, the moduli space of semi-stable sheaves, Hilbert scheme and Chow variety. Finally, he plans to study the Jacobian variety of line bundles on a non-reduced curve, which may reveal geometric properties of smooth curves by deformation and degeneration techniques. This project belongs to the subject of algebraic geometry, whose main objects are algebraic varieties defined by the solution sets of polynomial equations. Moduli spaces parameterize varieties of a given type. For instance, a donut and a car tire are of the same type, because they both have one hole, but a pretzel with three holes is different. An attractive aspect is that a moduli space for its objects tends itself to be a variety. Therefore, studying moduli spaces can help us understand the classification of algebraic varieties. In addition, moduli spaces have been extensively used in many other fields. The principal investigator expects that the outcome of this project can enrich the studies of other subjects, including combinatorics, dynamics, enumerative geometry and mathematical physics.