Special cubic zeros and the dual variety

Let F$F$ be a diagonal cubic form over Z$\mathbb {Z}$ in six 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$F$ in regions of diameter X ->infinity$X \rightarrow \infty$. Heath-Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general F$F$.