Singular Fano 4-folds

法诺三折已经被研究了近一个世纪，一维和二维的情况在19世纪得到解决。法诺三折没有分类，但已知它们只能被分为有限多个变形科。值得注意的是，对于一些重要的类已经实现了分类。特别是“著名的95”家族，其一般成员作为超曲面被多元反正则嵌入。Birkar关于Fano变异体有界性的著名定理证明了在任何维度上，在对可能奇点的非常一般的假设下，Fano变异体只有有限多个变形族。利用标准最小模型程序的终端奇异点，我们可以要求枚举某些子类。在4维中，准光滑的超曲面在2016年被分类。然而，与三维不同的是，对于四折，没有q-平滑结果，并且准光滑情况预计不会解释大多数超曲面。该项目将列举非准光滑的终端Fano 4-fold，从而首次表明准光滑假设的弱点。

Fano 3-folds have been studied for nearly a century, with the 1 and 2 dimensional cases being solved in the 19th century. There is no classification for Fano 3-folds, however it is known that they can be sorted into only finitely many deformation families. Notably, for some of the important classes classifications have been achieved. In particular, the "famous 95", the families whose general member is embedded pluri-anticanonically as a hypersurface.Birkar's celebrated theorem on the boundedness of Fano varieties proved that in any dimension, with very general hypotheses on the possible singularities, there are only finitely many deformation families for Fano varieties. Working with the terminal singularities of the standard minimal model program, we can ask to enumerate certain subclasses. In 4 dimensions, hypersurfaces that are quasismooth were classified in 2016. Unlike in 3 dimensions, however, there is no q-smoothing result for 4-folds, and the quasismooth case is not expected to account for the majority of hypersurfaces. The project will enumerate terminal Fano 4-folds that are not quasismooth giving the first indication of the weakness of the quasismooth assumption.