AI 中文总结
本文构造环面四维流形的形变以光滑化 Calabi-Yau 超曲面的奇点，提出逐面准则，并指出其覆盖 Batyrev--Kreuzer 普查中约 12.5% 的光滑化，其余需跨面关系。
AI 中文摘要
设 $\Delta$ 是一个具有原始边的自反四维多胞形。我们构造由其面扇定义的环面四维流形的形变，该形变在选定的二维面处光滑化一个一般反典范 Calabi-Yau 超曲面的奇点。该构造适用于面锥可光滑且对偶边具有内部格点的情况；其方程是 Cox 坐标中的三项式。当该面是唯一具有奇异锥的面时，它给出超曲面的光滑化。如果每个奇异面都满足这些条件，则在节点情形下可同时光滑化，并且一般而言，在半泛形变空间不可约的条件下也可同时光滑化。一个具有一个普通二重点且无光滑化的七顶点例子表明对偶边条件是尖锐的。对 Kreuzer--Skarke 分类的扫描表明，逐面准则解释了 Batyrev--Kreuzer 普查中 30,241 个光滑化中的 3,774 个；其余 26,467 个需要来自不同面的例外曲线类之间的关系。
英文摘要
Let $Δ$ be a reflexive four-polytope with primitive edges. We construct deformations of the toric fourfold defined by its face fan that smooth the singularities of a generic anticanonical Calabi-Yau hypersurface over a chosen two-dimensional face. The construction applies when the face cone is smoothable and the dual edge has an interior lattice point; its equations are trinomials in Cox coordinates. It gives a smoothing of the hypersurface when this is the only face whose cone is singular. If every singular face satisfies these conditions, simultaneous smoothing follows in the nodal case and, in general, under irreducibility of the semiuniversal deformation space. A seven-vertex example with one ordinary double point and no smoothing shows that the dual-edge condition is sharp. A scan of the Kreuzer--Skarke classification shows that the facewise criterion accounts for 3,774 of the 30,241 smoothings in the Batyrev--Kreuzer census; the other 26,467 require relations among exceptional curve classes from different faces.
Comments18 pages
