Laudal's Lemma in positive characteristic

Laudal's Lemma states that if $C$ is a curve of degree $d > s^2 + 1$ in $\mathbb P^3$ over an algebraically closed field of characteristic 0 such that its plane section is contained in an irreducible curve of degree s, then $C$ lies on a surface of degree $s$. We show that the same result does not hold in positive characteristic and we find different bounds $d > f(s)$ which ensure that $C$ is contained in a surface of degree $s$.

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