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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 $d$ ($d \ge 5$) surface $X$ in $\mathbb{P}^3$ for $d \ge e$ 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$) 3 added 10 characters in body Fix a Hilbert polynomial$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\$)

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