AI 中文总结
该研究证明了具有一个Puiseux对的有理单尖点曲线的自相交达到猜想上界，推导了相关递推恒等式与上界新公式，还得到奇点重数序列最后一项的公式，强化了文献结果的辛版本。
AI 中文摘要
对于任意(p,q)，我们证明了代数曲面中存在具有一个Puiseux对(p,q)的有理单尖点曲线，其自相交达到第一作者在文献[C]中猜想的上界m_{p,q}，从而强化了文献[C]结果的辛版本。这些“最优”曲线是通过研究两类无限族的有理双尖点曲线得到的，一类在CP²中，一类在CP¹×CP¹中。为便于计算，我们推导了某些递推恒等式，作为副产品得到了上界m_{p,q}的新公式，该公式更便于计算，且能让我们更好地理解该上界的性质。作为本研究的副产品，还得到了奇点重数序列最后一项的公式。
英文摘要
For any $(p,q)$, we proved the existence of a rational unicuspidal curve with one Puiseux pair $(p,q)$ in an algebraic surface, with a self-intersection which realizes the upper bound $m_{p,q}$ conjectured by the first-named author in \cite{C}, thus strengthening the symplectic version of the result in \cite{C}. These ``optimal" curves were obtained by examining two infinite families of rational bicuspidal curves, one in $CP^2$ and one in $CP^1\times CP^1$. In order to facilitate the computations, we derived certain recursive identities, and as a byproduct, we obtained a new formula for the bound $m_{p,q}$, which is more amenable to computations and gives us better insight concerning the nature of the bound. As a byproduct of this investigation, a formula for the last entry of the multiplicity sequence of the singularity was also found.
CommentsResults from a REU project, summer 2026. 10 pages. Comments Welcome!
