Show that if $Z$ is a curve of degree $b$ which contains $a\cdot b - 1$ points from intersection, also it contains the last one.

Question

Let $X, Y\subset \mathbb{P}^2$ of degree $a\geq 3$ and $b\geq 3$ such that #$(X\cap Y)=a\cdot b$.

Show that if $Z$ is a curve of degree $b$ which contains $a\cdot b - 1$ points from intersection, also it contains the last one.

Answer 1

Let me assume for simplicity that $X$ with $\deg X=a$ is smooth. Then the genus of $X$ is positive. If $Y\cap X=D, Z\cap X=E$, one gets $D-E$ is linearly equivalent to zero on $X$. But, by assumption, they have at least $ab-1$ points in common, so $D-E=P-Q$, where $P, Q$ are the remaining points. But, $P-Q$ is linearly equivalent to zero on a positive genus curve implies $P=Q$.