The Principal Investigator proposes to work on birational models of moduli spaces of curves. Specifically, he will continue his investigation of the relationships among the compactifications of the space of smooth abstract curves arising from the Minimal Model Program, Geometric Invariant Theory, and the deformation theory of curve singularities. In addition, he proposes to study moduli of curves together with maps to projective space; specifically, birational models of the Kontsevich space of stable maps arising from the Minimal Model Program, and also generalizations of Viscardi's construction of moduli spaces of stable maps with domains having singularities other than nodes.
An algebraic curve is, simply put, a polynomial in two variables; and the theory of algebraic curves is largely concerned with the relationship between the solutions of such a polynomial and the geometry of the set of solutions. To fully understand this relationship, we need to study the space of all possible curves 'the moduli space of curves' and its geometry. This in turn requires that we study different models of this space; and this is what the Principal Investigator proposes to do.
