Applications of Moduli Spaces of Maps of Nodal Curves

deJong/Vakil9970101 研究人员建议使用 Maxim Kontsevich 的“稳定映射”空间（最初由本世纪初的数学物理学激发）来解决各个领域的问题。 应用包括解释曲面上曲线的枚举几何中模形式的推测存在，理解各种组合问题（与球体分枝覆盖的几何形状有关），将稳定映射推广到正特征，以代数方式理解“相对”情况下稳定映射（“Gromov-Witteninvariants”）的原始动机，以及使用稳定映射来理解尖曲线的模空间。该提案涉及代数几何，研究由多项式方程定义的几何对象的数学领域。众所周知，节点代数曲线是代数几何中的强大工具。 这些曲线可以被认为是带有孔的曲面，其中成对的点“粘合在一起”。 本世纪初，理论物理学的思想导致了“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。 事实证明，这一发展非常富有成效，激发了各个领域的进步。 研究者的研究领域是节点代数曲线族在代数几何中的应用，特别是在其他领域的应用，如枚举几何、算术几何、组合学和数学物理。

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.