Space curves not contained in low degree surfaces in positive characteristic

Abstract

Let $C \subset \mathbf{P}^3$ be an integral projective curve not contained in a quadric surface. Set $d:=deg(C),g:=p_a(C),$ 

$\pi_1(d,3):= \left\{ $ $d^2/6 - d/2 + 1$ if $d/3 \in \mathbf{Z}$ 
                              $d^2/6 - d/2 + 1/3$ if $d/3 \notin \mathbf{Z}$ $\right$.

Here we prove in arbitrary characteristic that $g \le \pi_1(d,3)$ if $d \ge 25$.