Recently I've been trying to generalize the Cayley-Bacharach theorem to the case of quartics. Here's one version of what the theorem says for cubics:
Let $P_1,\dots,P_8$ be eight (distinct, closed) points in $\Bbb P^2$, with no four collinear and no seven on a conic. Then every cubic through these eight points also passes through a uniquely determined ninth point $P_9$.
The idea of the proof is that you show that these eight points impose independent conditions on cubics, i.e. the dimension of the vector space of cubic polynomials passing through $P_1,\dots,P_8$ is $\binom{3+2}{2}-8=2$ and so if $F,G$ are a basis for this vector space, then $V(F)\cap V(G)$ is nine points: $P_1,\dots,P_8$ and a ninth point which can be taken to be $P_9$ (it's easy to see that every curve in the pencil passes through $P_9$, since $aF+bG$ must be zero at $P_9$).
To show the independent conditions claim, one can construct reducible cubics through all sorts of configurations of seven points to show that the linear system of cubics through seven points has no other base points. This works for cubics because seven points can be decomposed as two points for a line and five points for a conic.
This seems to fail spectacularly for quartics, though. The claim would be that given 13 good enough points, all quartics through them also pass through three uniquely determined other points. If I wanted to use the same method of proof, I'd need to find conditions so that 13 points impose independent conditions on quartics, then I'd get a pencil and could conclude the same way. This claim about independent conditions would be the same as proving the linear system of quartics through twelve points has no other base points, and in general I don't think you can pick the same sort of reducible curves through special subsets of the points: a line is determined by two points, a conic by 5, a cubic by 9 (with some genericity conditions), and you can't get to 12 by combining these.
On the other hand, this is old enough that someone probably wrote down the answer somewhere before. I'm having real trouble finding it, though - I've located Eisenbud, Green, and Harris' Cayley-Bacharach Theorems and Conjectures which seems pretty cool but hasn't yielded an answer yet.
Is it known what conditions one must put on 13 points so that they impose independent conditions on quartics? If so, what are they, and if this is in the literature somewhere, could you tell me where?
Actually I think Theorem (CB5) from the Eisenbud, Green, Harris paper gives you exactly what you are asking about. At least it gives the following: Let C and D be plane quartics with no common component, and assume the scheme-theoretic intersection of C and D is reduced, let $\Gamma$ be their 16 points of intersection. Let $Z_1$ $\subset \Gamma$ be 13 of those points and let $Z_2$ be the remaining 3 points of $\Gamma$. Then CB5 says that the dimension of quartics containing $Z_1$ equals the dimension of quartics containing $\Gamma$ iff $Z_2$ imposes independent conditions on lines (ie iff the points of $Z_2$ are not colinear).
Edit: For a version of the above that also allows for non-reduced intersections of the two curves, refer to CB7 in the mentioned paper.