Applications of Moduli Spaces of Maps of Nodal Curves

deJong/Vakil 9970101研究者建议使用Maxim Kontsevich的“稳定映射”空间（最初是由本世纪初的数学物理学激发的）来解决各种领域的问题。 应用包括解释曲面上曲线的计数几何中模形式的存在性，理解各种组合问题（与球面的分歧覆盖的几何学有关），将稳定映射推广到正特征，代数地理解稳定映射的原始动机（“Gromov-Witteninvariants”）在“相对”的情况下，并使用稳定的映射来理解themoduli空间的尖曲线。这个建议涉及代数几何，研究由多项式方程定义的几何物体的数学领域。在代数几何中，节点代数曲线是一个强有力的工具。 这些曲线可以被认为是带有孔的表面，这些孔带有“粘在一起”的点对。 本世纪初，来自理论物理学的想法导致了“稳定映射”的引入，将某些类型的节点曲线映射参数化到另一个空间。 这一发展已被证明是非常富有成效的，引发了各个领域的进步。 研究者的研究领域是在代数几何中使用节点代数曲线族，特别是在其他领域的应用，如枚举几何、算术几何、组合数学和数学物理。

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.