Canonical curves with low apolarity

Let k be an algebraically closed field and let C be a nonhyperelliptic 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 Rg−3 in g − 2 variables. The apolarity ap(C) of C is the minimal number t of linear form in Rg−3 needed to write f as the sum of their divided power cubes. It is easy to see that ap(C) is at least g − 2 and P. De Poi and F. Zucconi classified curves with ap(C) = g −2 when k is the complex number field. In this paper, we give a complete, characteristic free, classification of curves C with apolarity g −1 (and g −2).