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Let $C_1 \cup C_2$ be a curve in $\mathbb{P}^3$ and $X$ be a smooth degree $d$ surface in $\mathbb{P}^3$ containing them and $d \ge 6$. Further, assume that the minimum degree polynomial in $I(C_1 \cup C_2)$ is of degree less than $d/2$. Is it true that there exists a smooth degree $d$ surface in $\mathbb{P}^3$ containing $C_1$ and a line? (The underlying field is always $\mathbb{C}$)

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$C_1$ and $C_2$ are both curves? $C_2$ is distinct from $C_1$? Is it any line, or a specific line? –  Will Sawin Jan 11 '13 at 23:20
    
@Sawin: $C_i$ are distinct curves. Answer to the last question is any line. –  Naga Venkata Jan 13 '13 at 13:25
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