I am reading a text by Prof. Szpiro Tata lectures on equations defining space curves. In the proof of Proposition $1.2$ on page $12$ he gives explicit description of the defining equations of a local complete intersection curve, say $C$ in $\mathbb{P}^3$. In the first step he produces a polynomial $F_1 \in H^0(\mathcal{I}_C(n))$, for some integer $n$, such that $C$ is an effective Cartier divisor in the surface defined by $F_1$. The question is: From the existence of the polynomial $F_1$ (given on page $13$) it is not clear if $n$ is some number or could be any "large enough" number. In particular, for any $n$ large enough, can we find a hypersurface in $\mathbb{P}^3$ of degree $n$ containing $C$ as an effective Cartier divisor?

The reason for asking this question is: I would have thought that the answer to the last question should be false if $C$ is not reduced since the adjunction formula (which exists for an effective Cartier divisor on any hypersurface in $\mathbb{P}^3$, a Gorenstein scheme) tells us that the arithmetic genus of a non-reduced curve/effective Cartier divisor depends on the degree of the hypersurface containing it as a Cartier divisor.