Canonical curves with low apolarity

الأصل · EN

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).