A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. Looijenga proved in 1981 that if a cusp singularity is smoothable, the minimal resolution of the dual cusp is the anticanonical divisor of some smooth rational surface. In 1983, the second author and Miranda gave a criterion for smoothability of a cusp singularity, in terms of the existence of a K-trivial semistable model for the central fiber of such a smoothing. We study these "Type III degenerations" of rational surfaces with an anticanonical divisor--their deformations, birational geometry, and monodromy. Looijenga's original paper also gave a description of the rational double point configurations to which a cusp singularity deforms, but only in the case where the resolution of the dual cusp has cycle length 5 or less. We generalize this classification to an arbitrary cusp singularity, giving an explicit construction of a semistable simultaneous resolution of such an adjacency. The main tools of the proof are (1) formulas for the monodromy of a Type III degeneration, (2) a construction via surgeries on integral-affine surfaces of a degeneration with prescribed monodromy, (3) surjectivity of the period map for Type III central fibers, and (4) a theorem of Shepherd-Barron producing the simultaneous contraction to the adjacency of the cusp singularity.

中文翻译：

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尖点奇点的平滑和有理双点邻接

尖点奇点是表面奇点，其最小分辨率是平滑有理曲线横向相遇的循环。尖点奇点自然是对偶的。Looijenga 在 1981 年证明，如果一个尖点奇点是光滑的，则双尖点的最小分辨率是某个光滑有理曲面的反正则除数。1983 年，第二作者和 Miranda 给出了一个尖点奇点的平滑性标准，根据这种平滑的中心纤维的 K-平凡半稳态模型的存在。我们研究这些带有反正则除数的有理曲面的“III 型退化”——它们的变形、双有理几何和单偶性。Looijenga 的原始论文还描述了尖点奇点变形的有理双点配置，但仅适用于双尖点分辨率的周期长度为 5 或更短的情况。我们将这种分类推广到任意尖点奇点，给出这种邻接的半稳定同时分辨率的显式构造。证明的主要工具是（1）III 型退化的单调公式，（2）通过手术在具有规定单调的退化的整体仿射表面上进行构造，（3）III 型周期图的满射性中心纤维，和 (4) Shepherd-Barron 定理产生对尖点奇点邻接的同时收缩。