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For simplicity, let us work over $\mathbb C$. The classical definition of gonality of a smooth curve $C$ is the mininal degree of a map $f: C \to \mathbb P^1$.

Now let $S$ be a smooth algebraic surface, $L$ be a divisor on $S$ such that $|L|$ is base point free and that $L^2>0$. Given any integer $d>0$, can we find a smooth curve $C$ in $|mL|$ for certain integer $m$ such that the gonality of $C$ is bigger than $d$?

Also if we have some similar notion of the gonality for higher dimensional varieties, we can ask the same question.

I think this should be true. But I really want to find a rigorous (and as simple as possible) proof for this statement.

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    $\begingroup$ I'm not sure your claim is true. The genus of a curve in $|mL|$ and the dimension of $|mL$ bot grow asymptotically as $m^2L^2/2$, so the image of $|mL|$ in the moduli space of curves has high codimension. $\endgroup$
    – rita
    Nov 22, 2012 at 12:53
  • $\begingroup$ Thanks for your comment, rita. There is a result saying that the smooth curve of degree $d$ in $\mathbb P^2$ has gonality $d−1$. This is an evidence for my statement to be true. $\endgroup$
    – Tong
    Nov 22, 2012 at 18:53

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