Intersections with elliptic curves on a K3 surface

Question

This is a fairly simple question.

Suppose $E$ is an elliptic curve on a K3 surface $X$. Can we say that $E$ must intersect any curve $D\subset X$ of genus $g(D)\geq3$ ?

Answer 1

In fact this is true for curves of genus at least 2.

For an elliptic curve $E$ in a $K3$ surface $X$, adjunction tells us that we have $E^2=0$. Then Riemann–Roch says that $h^0(X,E) \geq 2$; on the other hand, the ideal sheaf sequence for $E$ says that $h^0(X,E) \leq 2$. So in fact $h^0(X,E)=2$.

So the line bundle corresponding to $E$ has 2 linearly independent sections; since $E^2=0$ we conclude that the bundle is globally generated, so gives us a fibration $f:X \rightarrow \mathbf P^1$ with $E$ as one of the sections.

But then the only curves on $X$ which can be disjoint from $E$ are components of the fibres of $f$, which have genus 1 (if they are smooth fibres) or 0 (if they are components of reducible fibres).