Waring decompositions of special ternary forms with different Hilbert functions

TL;DR

This paper demonstrates the existence of specific ternary forms with equal Waring rank and apolar sets, but differing in Hilbert functions and regularity, using liaison theory and Cayley-Bacharach properties.

Contribution

It introduces new examples of ternary forms with identical Waring rank and apolar sets but distinct algebraic properties, expanding understanding of form classifications.

Findings

Existence of ternary forms with equal Waring rank and different Hilbert functions.

Application of liaison theory and Cayley-Bacharach properties to construct such forms.

Insights into the algebraic structure of special ternary forms.

Abstract

We prove the existence of ternary forms admitting apolar sets of points of cardinality equal to the Waring rank, but having different Hilbert function and different regularity. This is done exploiting liaison theory and Cayley-Bacharach properties for sets of points in the projective plane