Characterization of $d$-gonal curves on a K3 surface

Question

Let $X$ be a K3 surface and $C$ a curve on $X$. We say that $C$ is $d$-gonal if it admits a pencil of degree $d$ (and none of smaller degree).

I am wondering if there exist characterizations of $d$-gonal curves on $X$ for small $d$.

For example, if $d=2$ (that is, $C$ is hyperelliptic) then Saint-Donat proved that

1. either there exists an elliptic curve $E$ such that $C.E=2$; or
2. $C$ is linearly equivalent to $2B$, where $B$ is a curve of genus $2$.

(this is Theorem 5.2 in his famous thesis)

I am interested in particular for a result of the same flavor when $d=3$.

Answer 1

There is indeed such a characterisation. It is essentially due to Donagi-Morrison, see section 4 of the paper "Linear-systems on K3 Surfaces" in J. Diff. Geo. Let $C$ be a $d$-gonal curve on a K3 surface, and assume for simplicity that the $A \in W^1_d(C)$ achieves the Clifford index (this will be verified in generic cases). Then there is a line bundle $D$ on $X$ with special properties, e.g. $D$ restricted to $C$ attains the Clifford index and contains $A$ as a sub-line bundle (this restriction need not be $A$ but often will be). The line bundle $D$ has the numerical property $(C \cdot D)-(D \cdot D)-2=d-2$.

Two other closely related references you should look at are Green, Lazarsfeld "Special divisors on curves on a K3 surface" and Knutsen "On kth-order embeddings of K3 surfaces and Enriques Surfaces".