Minimal Model Program in Birational Geometry

代数几何是研究代数簇，即满足多项式方程组的点集。在过去的二十年里，代数几何在数学中发挥了核心作用，因为它在不同领域的许多应用。因此，对这些品种进行分类是至关重要的。由Mori在20世纪70年代开始的最小模型程序旨在将投影曲面的分类，即二维投影簇，推广到高维簇。这个程序在20世纪80年代成功地进行了三重投影，即三维投影簇。在高维空间中，关于射影簇几何的许多问题仍然是悬而未决的。本研究旨在解决其中的一些突出问题，在最小模型计划方面取得进展。

Algebraic Geometry is the study of algebraic varieties, i.e. the setof points satisfying a system of polynomial equations. In the last two decades, algebraic geometry has played a central role in Mathematics, due to its many applications in different fields. Thus, it is of crucial importance to find a classification of thesevarieties. The Minimal Model Program, started by Mori around the 1970's, aims to generalize the classification of projective surfaces,i.e. two dimensional projective varieties, to higher dimensional varieties. This Program was successfully carried out in the 1980's for projectivethree-folds, i.e. projective varieties of dimension three. In higher dimension, many problems concerning the geometry of projective varieties are still open. The proposed research aims to solve some of these outstanding problems, making progress in the Minimal Model Program.