A note on the self-intersection of rational unicuspidal curve with one Puiseux pair

Abstract

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 $C{P}^2$ and one in $C{P}^1\times C{P}^1$. 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.

Results from a REU project, summer 2026. 10 pages. Comments Welcome!