A footnote to a footnote to a paper of B. Segre

Abstract: The paper is devoted to a detailed study of sextics in three variables having a decomposition as a sum of nine powers of linear forms. This is the unique case of a Veronese image of the plane which, in the terminology introduced by Ciliberto and the first author in [12], is weakly defective, and non-identifiable. The title originates from a paper of 1981, where Arbarello and Cornalba state and prove a result on plane curves with preassigned singularities, which is relevant to extend the studies of B. Segre on special linear series on curves. We explore the apolar ideal of a sextic $F$ and the associated catalecticant maps, in order to determine the minimal decompositions. A particular attention is played to the postulation of the decompositions. Starting with forms with a decomposition $A$ of length $9$, the postulation of $A$ determines several loci in the $9$-secant of the $6$-Veronese image of $\mathbb P^2$, which include the lower secant varieties, and the ramification locus, where the decomposition is unique. We prove that equations of all these loci, including the $8$-th and the $7$-th secant varieties, are provided by minors of the catalecticant maps and by the invariant $H_{27}$ that we describe in Section 4.