Canonical curves with low apolarity

TL;DR

This paper classifies non-hyperelliptic smooth projective curves with low apolarity, specifically those with apolarity g-1 and g-2, extending previous complex field results to all characteristics.

Contribution

It provides a complete, characteristic-free classification of curves with apolarity g-1 and g-2, expanding the understanding of their algebraic and geometric properties.

Findings

Classified curves with apolarity g-1 and g-2 across all characteristics.

Extended previous complex field classifications to characteristic-free setting.

Connected apolarity to geometric properties of canonical curves.

Abstract

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 $g-2$).