CAREER: Algebraic Geometry of Moduli Spaces

职业：模空间的代数几何

• 批准号: 0134259
• 负责人: Brendan Hassett
• 金额: --
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0134259&HistoricalAwards=false
• 关键词: CAREER; Algebraic; Geometry; Moduli; Spaces

The investigator and his colleagueswill address a number offundamental problems aboutthe geometry and arithmetic of moduli spaces:What is the canonical model of the moduli space of curves of genus g?Can one construct natural spaces that interpolate between thecanonical model and the moduli space of stable curves?Can these be interpretted as moduli spaces in their own right?Can one count the number of rationalpoints of moduli spaces of curves, boundedwith respect to various heights? What are the naturalfunctions (effective divisors) one might use to specify theseheights?What are the naturally defined stratain the deformation space of a plane curve singularity?How can one interpret blow-ups along such strata?Algebraic geometry is the study of the solutionsto systems of polynomial equations. Geometricaspects of the solution sets translate into algebraicproperties of the equations, and vice versa. Thisapproach has the advantage that computers canmanipulate equations very efficiently. Promisinghypotheses can thus be checked on explicit examples.Increasingly, computational approaches to geometryare transforming the education of undergraduateand graduate students, both in pure mathematicsand in related fields where mathematics is applied.This emphasis on concrete examples and explicit computationcreates new research opportunities for young people,especially those who have only just started to learnthe technical intricacies of the subject.

调查员和他的同事将解决一些fundamental问题aboutthe几何和算术的模空间：什么是规范模型的模空间曲线的亏格g？能否构造出一个在规范模型和稳定曲线的模空间之间插值的自然空间？这些可以被解释为模空间本身吗？曲线模空间的有理点的数目，关于不同的高度有界吗？ 人们可以用什么样的自然函数（有效因子）来确定这些高度？在平面曲线奇点的变形空间中，自然定义的层是什么？如何解释沿着这样的地层的爆炸？代数几何是研究多项式方程组的解的学科。 解集的几何性质转化为方程的代数性质，反之亦然。 这种方法的优点是计算机可以非常有效地处理方程。 因此，有希望的假设可以在明确的例子上得到检验。越来越多的几何计算方法正在改变本科生和研究生的教育，无论是在纯几何学还是在应用数学的相关领域。这种对具体例子和明确计算的强调为年轻人创造了新的研究机会，特别是那些刚刚开始学习该学科技术复杂性的人。