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Fix a Hilbert polynomial $P$ of a non-plane curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of these curves is $e$. For a general curve $C$ in $Hilb_P$ (the Hilbert scheme corresponding to $P$) and a general smooth degree $d$ surface $X$ in $\mathbb{P}^3$ for $d \ge e$ containing $C$, should we expect that the dimension of the linear series $|C|$ which is equal to $h^0(\mathcal{O}_X(C))-1$ to be equal to zero? (Intuitively I would expect this because in this case if $C' \subset C$ irreducible then $C'^2<0$ implying $\dim |C'|=0$)

Fix a Hilbert polynomial $P$ of a non-plane curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of these curves is $e$. For a general curve $C$ in $Hilb_P$ (the Hilbert scheme corresponding to $P$) and a general smooth degree $d$ ($d \ge 5$) surface $X$ in $\mathbb{P}^3$ for $d \ge e+2$ containing $C$, should we expect that the dimension of the linear series $|C|$ which is equal to $h^0(\mathcal{O}_X(C))-1$ to be equal to zero? (Intuitively I would expect this because in this case if $C' \subset C$ irreducible then $C'^2<0$ implying $\dim |C'|=0$)

Fix a Hilbert polynomial $P$ of a curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of these curves is $e$. For a general curve $C$ in $Hilb_P$ (the Hilbert scheme corresponding to $P$) and a general smooth degree $d$ surface $X$ in $\mathbb{P}^3$ for $d \ge e$ containing $C$, should we expect that the dimension of the linear series $|C|$ which is equal to $h^0(\mathcal{O}_X(C))-1$ to be equal to zero? (Intuitively I would expect this because in this case if $C' \subset C$ irreducible then $C'^2<0$ implying $\dim |C'|=0$)

Fix a Hilbert polynomial $P$ of a non-plane curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of these curves is $e$. For a general curve $C$ in $Hilb_P$ (the Hilbert scheme corresponding to $P$) and a general smooth degree $d$ surface $X$ in $\mathbb{P}^3$ for $d \ge e$ containing $C$, should we expect that the dimension of the linear series $|C|$ which is equal to $h^0(\mathcal{O}_X(C))-1$ to be equal to zero? (Intuitively I would expect this because in this case if $C' \subset C$ irreducible then $C'^2<0$ implying $\dim |C'|=0$)