F-理论中SU（5）-模型的小分辨
Small resolutions of SU(5)-models in F-theory
摘要：
我们提供了一个明确的desingularization和研究所产生的纤维几何椭圆纤维的四重定义的维尔斯特拉斯模型承认分裂A_4奇异性的一个除数的判别轨迹。这些变体被用于F理论中的SU（5）大统一理论的几何工程。去奇异化是由奇异点的小分辨率给出的。在基底中分解余维1的奇异性后，I_5纤维自然出现。剩余的更高的余维奇点，然后美丽地描述了一个四维仿射二项式品种，导致六个不同的小决议的椭圆纤维四倍。这六个小分辨率定义了不同的四重，它们通过形成二面角群的翻转过渡网络相互连接。这些外来纤维在基地的位置被映射到conifold点的threefolds定义的IIB型orientifold限制的F-理论。全分辨率具有有趣的性质，特别是对于余维3的纤维：奇异纤维的秩不一定增加，纤维不一定在Kodaira列表中，有些甚至不是（扩展）Dynkin图。
英文摘要：
We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered fourfolds defined by Weierstrass models admitting a split A_4 singularity over a divisor of the discriminant locus. Such varieties are used to geometrically engineer SU(5) Grand Unified Theories in F-theory. The desingularization is given by a small resolution of singularities. The I_5 fiber naturally appears after resolving the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four dimensional affine binomial variety which leads to six different small resolutions of the the elliptically fibered fourfold. These six small resolutions define distinct fourfolds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base is mapped to conifold points of the threefolds that defines the type IIB orientifold limit of the F-theory. The full resolution have interesting properties, specially for fibers in codimension three: the rank of the singular fiber does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagram.
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