Rational Points on Algebraic Varieties and Geometry of Curves and Surfaces

这个项目研究了代数几何和数论中的几个基本问题。第一部分是关于代数方程解的几何。设X是定义在数域K上的簇。是否存在有限扩张L/K使得X(L)是稠密的？人们根据X的几何形状来寻找何时发生这种情况的标准，但这种标准应该采取什么形式仍然不清楚。第二部分是关于平面曲线的几何问题。抽象曲线什么时候是光滑平面曲线的退化？这种极限平面曲线的几何性质有哪些共同之处？第三部分涉及曲面奇异性的分类。给定一族射影正规曲面族，这个族的不变量什么时候是不变的？具体地说，如何判断正则因子的自交是否为常量？最后一部分涉及为固定的小代数的曲线寻找明确的参数化法。代数几何学家研究多项式方程的解的结构。自古以来，圆形和椭圆形等几何图形就让建筑师、科学家和艺术家着迷。表示这样一个数字的最简洁的方法是把它描述为一个方程的解。一个基本问题是对可以用给定形式的方程表示的抽象图形进行分类。另一种是理解给定形式的所有方程的解的共同几何性质。本项目旨在为特殊类型的曲线和曲面回答这些问题。

This project addresses several fundamental questions inalgebraic geometry and number theory.The first part concerns the geometry of solutions toalgebraic equations.Let X be a variety defined over a number field K.Does there exist a finite extension L/K such thatX(L) is dense? One seeks criteria for when thisoccurs in terms of the geometry of X.It is still not clear what form such criteria should take. The second part concerns the geometry of plane curves.When is an abstract curve a degenerationof smooth plane curves? What geometric properties dosuch limiting plane curves have in common?The third part involves the classification of surfacesingularities. Given a family of projective normalsurfaces, when are the invariants of such a familyconstant? Specifically, how does one tell whetherthe self-intersection of the canonical divisor is constant? The last part involves finding explicitparametrizations for the curves of a fixed small genus.Algebraic geometers study the structure of solutions topolynomial equations. From ancient times, geometricfigures like circles and ellipses have fascinated architects, scientists, and artists. The most compactway to represent such a figure is to describe it as the solutions to an equation. One basic question is to classify the abstract figures that may be representedby equations of a given form. Another is to understandthe common geometric properties of the solutions toall the equations of a given form. This project seeksto answer these questions for special types of curves and surfaces.