Coset intersection of irreducible plane cubics

In a projective plane $PG(2,mathbb K )$ over an algebraically closed field $mathbb K $ of characteristic $pge 0$ , let $Omega $ be a pointset of size $n$ with $5le n le 9$ . The coset intersection problem relative to $Omega $ is to determine the family $mathbf F$ of irreducible cubics in $PG(2,mathbb K )$ for which $Omega $ is a common coset of a subgroup of the additive group $(mathcal F ,+)$ for every $mathcal F in mathbf F$ . In this paper, a complete solution of this problem is given.