I am wondering wether the action of the Weyl group $W\_X$ of a K3 surface $X$ is transitive on the sets of curves of fixed genus. 


Suppose $W\_X$ is non-trivial. Given two curves $C,C'$ of genus $g\geq2$ on $X$, does there exists an element $\sigma$ of the Weyl group such that $\sigma C =C'$ ?



Weyl group of a K3 surface
Except in trivial cases, this is never true. The closure of the Kähler cone of $X$ is a fundamental domain for $W\_X$ (see Barth et al., *Compact complex surfaces*, ch. VIII, Prop. 3.10)). Thus if your curves $C$ and $C'$ are ample (which just means that they meet all $(-2)$-curves), they are not conjugate under $W\_X$. 


