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, Macaulayʼs 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 ℓ1,…,ℓt∈Rg−3 needed to write f as the sum of their divided power cubes.It is easy to see that ap(C)⩾g−2 and P. De Poi and F. Zucconi classified curves with ap(C)=g−2 when k≅C. In this paper, we give a complete, characteristic free, classification of curves C with apolarity g−1 (and g−2).