A Footnote to a Footnote to a Paper of B. Segre

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 indeed the unique case of a Veronese image X of the plane which, in the terminology introduced by Ciliberto and the first author, is {weakly defective}, and non-identifiable: a general sextic of the 9-secant variety of X has two minimal decompositions. The title originates from a famous 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. The result (Theorem 3.2) says that the linear system of sextics with 9 general nodes in P^2 is the unique non-superabundant system of plane curves with general nodes whose (unique) member is non-reduced.