  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 to associate to $C$ a cubic form $f$ in the divided power $k$--algebra
$R$ in $g-2$ variables. The apolarity of $C$ is the minimal number $t$ of
linear form in $R$ needed to write $f$ as sum of their divided power cubes.
  It is easy to see that the apolarity of $C$ is at least $g-2$ and P. De Poi
and F. Zucconi classified curves with apolarity $g-2$ when $k$ is the complex
field. In this paper, we give a complete, characteristic free, classification
of curves $C$ with apolarity $g-1$ (and $g-2$).
