Semistable Reduction Problems, and Uniformity Problems in Arithmetic Geometry

在这个项目中，主要研究者将研究半稳定约简问题，利用de Jong的变换方法，Abramovich和de Jong的Torifying Blowup，以及Kempf， Knudsen， Mumford和Saint-Donat的环面几何。他将把他关于纤维幂的结果推广到对数情况，并利用Alexeev和Nakamura的紧化模空间继续研究阿贝尔变异上的稳定积分点。这个项目属于算术几何的一般领域，这个学科融合了数学中两个最古老的领域：数论和几何。事实证明，这种结合非常富有成效，最近解决了几代人经受住的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机和光盘都是必不可少的。

Abramovich 9700520 In this project, the principal investigator will study semistable reduction problems, making use of the Alteration method of de Jong, the Torifying Blowup of Abramovich and de Jong, and Toroidal Geometry of Kempf, Knudsen, Mumford and Saint-Donat. He will work on generalizing his results on fibered powers to the logarithmic case, and continue the study of stably integral points on abelian varieties, using the compactified moduli spaces of Alexeev and Nakamura. This project falls into the general area of arithmetic geometry, a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers and compact disks.