On A_k-singularity on a plane curve of fixed degree

Abstract

Let $k(d)$ be the maximal possible integer $k$ such that there exists a plane curve of degree $d$ with an $A_k$--singularity. We construct a plane curve of degree $28s+9$ ($s\in\Z_{\ge 0}$) which has an $A_k$--singularity with $k=420s^2+269s+42$. Therefore one has $\underline{\lim}_{d\to\infty}k(d)/d^2\ge 15/28$ (pay attention that $15/28>1/2$).