Special cubic zeros and the dual variety

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 $F$.