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 $A$.


1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$ 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. $\endgroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.