On $k$-sheeted covering of curves

Question

Let $C \subset \mathbb P^3$ be a smooth complete intersection curve given by $2$ hypersurfaces of degree at least $4$ in $\mathbb P^3$. Then can it happen that:

1. $C$ is a $2$-sheeted covering of a curve of low genus (say $\leq 3$)?

2. $C$ is a $3$-sheeted covering of an elliptic curve?

Note that, in both the cases by Riemann–Hurwitz theorem we get a value of the degree of the ramification divisor. So the question boils down to whether there really exist ramification divisors having degrees those values.

Are there any other ways to give a bound on the degree of the ramification divisor in these two situations which contradicts the previously obtained value?

Any indication on whether this can happen (or can't happen) is appreciated.

Answer 1

Edit: As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: suppose $C$ is a $(p,q)$ complete intersection, with $p\geq q$. The maximum number $\ell$ of points of $C$ lying on a line is $\leq p\ $ — otherwise the line is contained in $C$. By B. Basili, Indice de Clifford des intersections complètes de l’espace, Bull. S. M. F. 124, no. 1 (1996), p. 61-95, the gonality of $C$ is $pq-\ell$, hence $\geq p(q-1)$. But if $C$ is a 2-sheeted cover of a curve $D$ of genus $\leq 3$, a $g^{1}_3$ on $D$ lifts to a $g^1_6$ on $C$; similarly if $C$ is a 3-sheeted cover of an elliptic (or more generally hyperelliptic) curve $E$, a $g^1_2$ on $E$ lifts to a $g^1_6$ on $C$. Hence the gonality of $C$ is $\leq 6$, which is less than $p(q-1)$ if $q\geq 4$.