Special cubic zeros and the dual variety

TL;DR

This paper investigates special zeros of diagonal cubic forms in six variables using a dual variety approach within the delta method, providing new insights and an axiomatic framework for broader classes of forms.

Contribution

It introduces a novel analysis of the dual variety in the delta method for six-variable cubic forms, extending previous work in four variables and proposing a general axiomatic framework.

Findings

Unconditional weighted count of special zeros in large regions.

New features captured in the analysis differ from prior four-variable results.

Framework applicable to more general forms beyond the specific case.

Abstract

Let $F$ be a diagonal cubic form over $\mathbb{Z}$ in $6$ variables. From the dual variety in the delta method of Duke--Friedlander--Iwaniec and Heath-Brown, we unconditionally extract a weighted count of certain special integral zeros of $F$ in regions of diameter $X \to \infty$. Heath-Brown did the same in $4$ variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general $F$.