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Let $X$ be a K3 surface. Let $L$ be an ample line bundle on $X$. When/how can we say that any smooth curve $C\in |L|$ has maximal gonality $k=[\frac{g+3}{2}]$ and Clifford dimension 1. Is there some criterion to check if the ample linear system satisfies that? Thanks!

$\textbf{Edit:}$ I am interested in the specific case when $X$ is the Kummer surface associated to an abelian surface $A$.

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This is the case when $\mathrm{Pic}(X)=\mathbb{Z}[L]$, as a consequence of two results: the theorem of Lazarsfeld (Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299-307) which states that a general curve in $|L|$ is Brill-Noether general, hence satisfies your requirement, and the theorem of Green-Lazarsfeld (Special divisors on curves on a K3 surface, Invent. Math. 89 (1987), no. 2, 357-370) which says that all smooth curves in $|L|$ have the same Clifford index.

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  • $\begingroup$ Thanks @abx! When $X$ is the Kummer surface associated to an abelian surface, then can this happen? That is the specific case I am looking at. Maybe I should edit the question. $\endgroup$
    – gradstudent
    Commented Oct 19, 2015 at 10:06
  • $\begingroup$ Unfortunately no -- a Kummer surface has a very high Picard number (>16). $\endgroup$
    – abx
    Commented Oct 19, 2015 at 10:30
  • $\begingroup$ Doesn't Lazarsfeld actually prove that any smooth curve in $|L|$ is Brill-Noether general (Corollary 1.4 in his paper)? $\endgroup$
    – dhy
    Commented Oct 19, 2015 at 18:59
  • $\begingroup$ @dhy, In the paper, Lazarsfeld assumes that every curve $C\in |L|$ is reduced and irreducible. $\endgroup$
    – gradstudent
    Commented Oct 23, 2015 at 18:36
  • $\begingroup$ @poorna But that follows from Pic(X)=Z[L]; I think as long as that's true, it doesn't matter if C is generic in the linear system. $\endgroup$
    – dhy
    Commented Oct 23, 2015 at 19:00

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