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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 $\sigma C =C'$ ?

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  • $\begingroup$ Certainly not in general: there are K3 surfaces with Picard number >1 containing no $(-2)$-curves. $\endgroup$
    – abx
    Commented Feb 8, 2014 at 14:55
  • $\begingroup$ ok sorry, let me edit $\endgroup$
    – Heitor
    Commented Feb 8, 2014 at 14:59

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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 $W_X$.

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    $\begingroup$ Still no. Actually the ampleness condition is superfluous: your curves are always nef, i.e. in the closure of the ample cone. Thus the above argument still applies. $\endgroup$
    – abx
    Commented Feb 8, 2014 at 17:28

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