Counting plane cubic curves over finite fields with a prescribed number of rational intersection points

For each integer $$k \in [0,9]$$ , we count the number of plane cubic curves defined over a finite field $${{\mathbb {F}}}_q$$ that do not share a common component and intersect in exactly $$k\ {{\mathbb {F}}}_q$$ -rational points. We set this up as a problem about a weight enumerator of a certain projective Reed–Muller code. The main inputs to the proof include counting pairs of cubic curves that do share a common component, counting configurations of points that fail to impose independent conditions on cubics, and a variation of the MacWilliams theorem from coding theory.