A problem in Hartshorne (Chapter IV, Ex. 6.2) claims that a rational curve $X$ of degree $5$ in $\mathbb{P^3}$ is always contained in a cubic surface.
However, let $X$ be a nonsingular curve of type $(1,4)$ on a nonsingular quadric surface $Q$ in $\mathbb{P}^3$. Then $X$ has degree $5$ and genus $0$. Suppose that $X$ is contained in a cubic surface $S$. Then $Q \cap S$ is a divisor $D$ of type $(3,3)$ on $Q$, and $X$ is a component of $D$. The difference (of divisors) $D - X$ must be both effective and of type $(2,-1)$, an impossibility.
What is wrong?