Canonical curves with low apolarity

Let $k$ be an algebraically closed field and let $C$ be a non--hyperelliptic smooth projective curve of genus $g$ defined over $k$. Since the canonical model of $C$ is arithmetically Gorenstein, Macaulay's theory of inverse systems allows us to associate to $C$ a cubic form $f$ in the divided power $k$--algebra $R^{g-3}$ in $g-2$ variables. The apolarity $\ap(C)$ of $C$ is the minimal number $t$ of linear form $\ell_1,\dots,\ell_t\in R^{g-3}$ needed to write $f$ as sum of their divided power cubes. It is easy to see that $\ap(C)\ge g-2$ and P\. De Poi and F\. Zucconi classified curves with $\ap(C)=g-2$ when $k\cong\C$. In this paper, we give a complete, characteristic free, classification of curves $C$ with apolarity $g-1$ (and $g-2$).