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.
