Inverting catalecticants of ternary quartics

TL;DR

This paper investigates the geometric and algebraic properties of the reciprocal variety associated with catalecticant matrices of ternary quartics, revealing key invariants and their geometric interpretations.

Contribution

It provides the degree and ML-degree of the reciprocal variety, explains the inequality between these invariants, and characterizes the locus contributing to the degree.

Findings

Degree of reciprocal variety is 85.

ML-degree of the LSSM is 36.

Only the rank-1 locus contributes to the degree.

Abstract

We study the reciprocal variety to the linear space of symmetric matrices (LSSM) of catalecticant matrices associated with ternary quartics. With numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of the LSSM. We provide a geometric explanation to why equality between these two invariants is not reached, as opposed to the case of binary forms, by describing the intersection of the reciprocal variety and the orthogonal of the LSSM in the rank loci. Moreover, we prove that only the rank-$1$ locus, namely the Veronese surface $\nu_4(\mathbb{P}^2)$, contributes to the degree of the reciprocal variety.