Let $P_1,\ldots,P_9$ be $9$ different points in a cubic $C$. Let $S$ be a sextic passing through $P_1,\ldots,P_8$ with multiplicity at least $2$ and passing through $P_9$ with multiplicity at least $1$. I have read that in this case $S$ passes through $P_9$ with multiplicity at least $2$, but I do not really know why.
It looks like some kind of generalization of Cayley-Bacharach Theorem:
If a cubic passes through $8$ of the $9$ points of intersection of two cubics, then it passes through the ninth.
Nevertheless, it is claimed to be a consequence of Abel's Theorem:
Let $X$ be a compact Riemann surface. Let $D$ be a divisor of degree $0$ on $X$. Then $D$ is the divisor of a meromorphic function iff it is zero under the Abel-Jacobi map.
I have not been able to relate these results.
By the way, we are allowed to impose any condition on the points $P_1,\ldots,P_9$.
If the 9 points, $P_1,\ldots,P_9$ are the intersection of $C$ with another cubic $D$, then $2D$ gives you the divisor $\sum 2P_i$. If $S$ gives you $2P_1+\cdots +2P_8+P_9+P_{10}$, then these two divisors are linearly equivalent and thus $P_9\sim P_{10}$. On a cubic, this means $P_1=P_{10}$.