Applications of Moduli Spaces of Maps of Nodal Curves

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deJong/Vakil9970101The investigator proposes to use Maxim Kontsevich's space of "stable maps"(originally motivated by mathematical physics earlier this decade) totackle problems in a variety of fields. Applications includeexplaining the conjectured presence of modular forms in the enumerativegeometry of curves on surfaces, understanding various combinatorialproblems (relating to the geometry of ramified covers of the sphere),generalizing stable maps to positive characteristic, algebraicallyunderstanding the original motivation for stable maps ("Gromov-Witteninvariants") in a "relative" case, and using stable maps to understand themoduli space of pointed curves.This proposal concerns algebraic geometry, an area of mathematics studying geometric objects defined by polynomial equations. It has long been known that nodal algebraic curves are a powerful tool in algebraic geometry. These curves can be thought of as surfaces with holes with pairs of points "glued together". Earlier this decade, ideas from theoretical physics led to the introduction of "stable maps", parametrizing certain kinds of maps of nodal curves into another space. This development has proved to be very fruitful, sparking advances in a variety of fields. The investigator'sarea of research is the use of families of nodal algebraic curves inalgebraic geometry, in particular with applications to other fields suchas enumerative geometry, arithmetic geometry, combinatorics, andmathematical physics.