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Let $F$ be a diagonal cubic form over $\mathbb{Z}$ in $6$ variables. From thedual variety in the delta method of Duke--Friedlander--Iwaniec and Heath-Brown,we unconditionally extract a weighted count of certain special integral zerosof $F$ 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 alsoput forth an axiomatic framework for more general $F$.

Special cubic zeros and the dual variety