The uniqueness of a fixed degree singular plane model

Abstract

Let $X$ be the normalization of an integral degree $d\ge 9$ plane curve $Y$. We prove that $X$ has a unique $g^2_d$ if $h^1(\mathbb{P}^2,\mathcal{I}_Z(\lceil d/2\rceil -3))=0$, where $Z$ is the conductor of $Y$. Moreover, $Y$ is the unique plane model of $X$ of degree at most $d$.