This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The moduli space of stable curves gives a beautiful compactification of the moduli space of compact Riemann surfaces, and is one of the most studied objects in algebraic geometry. The PI will investigate alternate compactifications, with the long-term goal of constructing a modular interpretation for the canonical model of the moduli space. The first objective is to give a systematic classification of all classes of singular curves which are deformation-open and satisfy the unique limit property - any such class gives rise to a modular compactification. The second objective is to study these alternate compactifications from the perspective of the minimal model program, factoring the birational maps between them into divisorial contractions and flips, and studying the numerical properties of the canonical class at every intermediate stage.
Quite generally, the theory of moduli is concerned with the underlying numerical parameters of a given class of geometric objects. For example, a circle is determined by a single modulus (its radius), while a triangle has three moduli (its three side lengths). Furthermore, the moduli associated to a geometric object typically satisfy fundamental constraints. For example, the fact that the sum of any two side-lengths of a triangle must be less than the third constrains the possible moduli parameters of a triangle. For modern geometry and physics, Riemann surfaces comprise one of the most important classes of geometric objects - their moduli are complex numbers and the relations constraining these moduli remain deeply mysterious. By viewing the moduli parameters of a Riemann surface as solutions to a system of polynomial equations, the PI will investigate these relations through the techniques of higher-dimensional algebraic geometry.
