Applications of Moduli Spaces of Maps of Nodal Curves

Dejong/Vakil9970101这位研究人员提议使用马克西姆·康采维奇的“稳定地图”空间(最初是由本世纪初的数学物理推动的)来解决各种领域的问题。应用包括解释曲面上曲线的枚举几何中模形式的猜想的存在，理解各种组合问题(与球面的分支覆盖的几何有关)，将稳定映射推广为正特征，从代数上理解相对情况下稳定映射(Gromov-Wittenant不变量)的原始动机，以及使用稳定映射来理解点曲线的模空间。这一建议涉及代数几何，这是一个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一个强有力的工具。这些曲线可以被认为是带有孔洞的曲面，这些孔洞的点成对地粘合在一起。本世纪初，来自理论物理的想法导致了“稳定映射”的引入，将某些类型的节点曲线映射到另一个空间。事实证明，这一发展非常富有成效，在各个领域引发了进展。研究人员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，如计数几何、算术几何、组合学和数学物理中的应用。

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.