On equations defining space curves 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.
 A: Let $\ell$ be the line in $\mathbb{P}^{3}$ cut out by the ideal $I=(x,y).$  It is true that if you look at the subscheme $\overline{\ell}$ of $\mathbb{P}^{3}$ cut out by $I^2$ (which is saturated) and its induced subschemes of smooth surfaces of different degrees containing $\ell,$ you will get nonreduced curves of different arithmetic genera.  However, $\overline{\ell} \subseteq \mathbb{P}^{3}$ is not a local complete intersection.
EDIT 1:  The number $n$ in Szpiro's proof is a means to the end of finding 4 generators for the homogeneous ideal of a given equidimensional lci curve in $\mathbb{P}^{3}.$  Picking a different $n$ may result in a different Cartier divisor on a different surface, and down the line this may result in a different collection of 4 generators, but this is OK if we fix $n$ once and for all in the "Existence of $F_1$" step. 
EDIT 2:  What is more problematic is the discussion at the top of p.14 in which the number $n$ is repurposed.  However, I think this can be fixed if instead of $n$ we use $n+n'$ for some sufficiently large $n'.$
