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

TL;DR

This paper investigates the unique decompositions of certain sextic forms in three variables, analyzing their geometric and algebraic properties, and identifying loci related to secant varieties using catalecticant maps.

Contribution

It provides a detailed study of the apolar ideal and catalecticant maps for sextic forms, characterizing loci of decompositions and secant varieties, including equations and invariants.

Findings

Characterization of loci in the 9-secant of the Veronese image

Equations of secant varieties via minors of catalecticant maps

Identification of an invariant H_{27} related to decompositions

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.