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.