具有尖点奇点的曲线的对数 MMP 约束
Log MMP Constraints on Curves with Cuspidal Singularities
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中文总结 AI 辅助
该研究运用对数极小模型程序证明陈维民提出的单尖有理曲线自交数上界，推广到任意尖点有理曲线并与辛界比较，还证明不存在特定次数、亏格及尖点的平面曲线，回答了相关问题。
中文摘要 AI 辅助
设 X 为光滑射影曲面，C⊂X 为具有一个 Puiseux 对 (p,q) 的单尖有理曲线。我们运用对数极小模型程序（log minimal model program），证明了由陈维民（Weimin Chen）在研究接触结构及其辛填充过程中提出的关于 C 的自交数的精确上界，该上界对应于文献[C, Conjecture 1.15]。随后，我们将该论证推广到具有任意尖点奇点的有理曲线，并将得到的代数界与 Golla 和 Starkston 的辛界（文献[GS]）进行比较。最后，我们证明不存在次数为 102、亏格为 10 且具有 (36,289) 尖点的平面曲线，从而回答了 Evans 在文献[E, Remark 7.3.5]中提出的、关于与 McDuff-Schlenk 阶梯的最终后斐波那契步相关的代数曲线的问题。
英文摘要
Let $X$ be a smooth projective surface and let $C\subset X$ be a unicuspidal rational curve with one Puiseux pair $(p,q)$. We use the log minimal model program to prove a sharp upper bound on the self-intersection of $C$ conjectured by Weimin Chen in [C, Conjecture 1.15] in the course of his study of contact structures and their symplectic fillings. We then extend this argument to rational curves with arbitrary cuspidal singularities and compare the resulting algebraic bounds with the symplectic bounds of Golla and Starkston [GS]. Finally, we prove that there is no plane curve of degree $102$ and genus $10$ with a $(36,289)$-cusp, answering a question of Evans [E, Remark 7.3.5] about algebraic curves associated with the final post-Fibonacci step of the McDuff-Schlenk staircase.
