Geometry of Moduli Spaces, Geometric Invariant Theory, and Deformations of Singularities

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This proposal is concerned with the study of moduli spaces in algebraic geometry. By using methods of Geometric Invariant Theory, deformation theory, and enumerative geometry, the investigator will pursue several projects in order to elucidate the geometry of well-studied moduli spaces and to adopt new approaches to less studied ones. In the first project, the investigator will continue the study of finite Hilbert stability of embedded curves with the goal of advancing the log minimal model program for the Deligne-Mumford moduli space of stable curves. In the second project, the investigator will approach fundamental open questions about effective and ample divisors on this moduli space and its variants. The study of singularities is an important ingredient of these two projects. In the third project, the investigator will extend recent results in the theory of curves to certain classes of higher-dimensional varieties (e.g., K3 surfaces) and their moduli spaces.An algebraic variety is a collection of solutions to a system of polynomial equations. Algebraic varieties are fundamental objects of study in mathematics and, in particular, in the field of algebraic geometry, to which this proposal belongs. Variation of the polynomials' coefficients gives rise to a moduli space for a given class of varieties. The study of moduli spaces is essential to understanding algebraic varieties themselves and, ultimately, to solving systems of polynomial equations. The investigator proposes to study moduli spaces of algebraic varieties depending on one or two free parameters using both classical and modern techniques. The broader impacts of the proposal include advancing an active research program and co-organizing workshops on moduli spaces and related problems.