AI 中文总结
本文分类了590个自反四维多胞体，其反典范超曲面至多含孤立奇点，发现唯一不可光滑化镜像对（Hodge数(20,26)与(26,20)），并指出自对偶24胞体是唯一双方光滑的成员。
AI 中文摘要
我们对自反四维多胞体进行分类，使得与该多胞体及其极多胞体的面扇相关的通用反典范超曲面至多具有孤立奇点。在Kreuzer-Skarke分类中，恰好有590个这样的多胞体（在格等价意义下）。它们的二维面属于十二个格同构类，全部为三角形、区域多胞体或自反多边形。唯一不可光滑化的局部模型是循环商 $\frac13(1,1,1)$ 和 $\frac15(1,1,3)$，以及第一Hirzebruch曲面上的反典范锥；没有奇点具有无光滑化分量的正维约化形变空间。在此分类中，Hirzebruch锥出现在唯一的镜像对中，对应于具有22和26个顶点的多胞体。我们明确展示这一对：一个超曲面不可光滑化，而其镜像的所有奇点局部可光滑化。其crepant分辨的Hodge数为 $(20,26)$ 和 $(26,20)$。在相反的极端情况下，自对偶的24胞体是该分类中唯一在crepant分辨前两个超曲面都光滑的成员。
英文摘要
We classify reflexive four-polytopes for which the generic anticanonical hypersurfaces associated with the face fans of the polytope and its polar both have at most isolated singularities. There are exactly 590 such polytopes up to lattice equivalence in the Kreuzer-Skarke classification. Their two-dimensional faces belong to twelve lattice-isomorphism classes, all triangles, zonotopes, or reflexive polygons. The only nonsmoothable local models are the cyclic quotients $\frac13(1,1,1)$ and $\frac15(1,1,3)$ and the anticanonical cone over the first Hirzebruch surface; no singularity has a positive-dimensional reduced deformation space without a smoothing component. Within this classification, the Hirzebruch cone occurs in a unique mirror pair, associated with polytopes having 22 and 26 vertices. We exhibit this pair explicitly: one hypersurface is nonsmoothable, while all singularities of its mirror are locally smoothable. The crepant resolutions have Hodge numbers $(20,26)$ and $(26,20)$. At the opposite extreme, the self-dual 24-cell is the unique member of the classification for which both hypersurfaces are smooth before crepant resolution.
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