Rational Points on Algebraic Varieties and Geometry of Curves and Surfaces

这个项目解决了代数几何和数论中的几个基本问题。第一部分涉及代数方程解的几何。设X是定义在数域K上的一个变种。是否存在一个有限的扩展L/K使得X(L)是稠密的？人们在x的几何中寻找这种情况何时发生的标准，但仍然不清楚这种标准应该采取什么形式。第二部分涉及平面曲线的几何。什么时候抽象曲线是光滑平面曲线的退化？这些极限平面曲线有什么共同的几何性质？第三部分涉及曲面奇异点的分类。给定一组射影法曲面，该族常数的不变量是什么？具体来说，如何判断正则除数的自交是否为常数？最后一部分涉及寻找固定小格曲线的显式参数化。代数几何学者研究拓扑多项式方程解的结构。自古以来，圆形和椭圆等几何图形就吸引着建筑师、科学家和艺术家。表示这种图形的最简洁的方法是将其描述为一个方程的解。一个基本问题是对可能由给定形式的方程表示的抽象图形进行分类。另一个是理解所有给定形式方程解的共同几何性质。这个项目试图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.