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Let $C$ be a reduced (not irreducible) projective curve of degree $d$ such that $C$ contains at most double points. By a result due to Kleiman and Altman, we know that there exists a smooth surface containing $C$. Can we compute $n$ (explicitly) such that for any $m \ge n$, there exists a smooth degree $m$ surface containing $C$? What happens if we assume that $C$ is smooth?

I was trying to solve this question in the following way: If I understand correctly, Theorem $7$ in "Bertini theorems for hypersurface sections containing a subscheme" by Kleiman and Altman, states that $C$ can be embedded in a hypersurface in $\mathbb{P}^3$ (because if the "embedded dimension" $e(C)$ is equal to $1$ (resp. $2$) then the dimension of $X_e$ is equal to $1$ (resp. $0$) by definition) and also gives a rough estimate for the value of $n$ stated in the question. But there should be an error in my understanding because if I take $C$ to be of the form $\cup C_i$ where $C_i$ are $-2$ exceptional curves and $C_i.C_j \le 1$, then it cannot be embedded into a smooth hypersurface in $\mathbb{P}^3$ of degree not equal to $4$ (due to contradiction with the adjunction formula). Could someone point out precisely, where I am getting it wrong?

Note $X_e$ consists of the set of points on $C$ with embedded dimension $e$.

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  • $\begingroup$ The condition you wrote down is different from the minimum degree of a smooth surface. $\endgroup$
    – Will Sawin
    Feb 10, 2014 at 22:45
  • $\begingroup$ @Sawin: You are right. I would appreciate an answer to either of the question. I am more interested in the second question. $\endgroup$
    – user43198
    Feb 10, 2014 at 23:03
  • $\begingroup$ What is your definition of "minimum degree"? Certainly every such curve $C$ is contained in a smooth, proper surface that admits a very ample line bundle. However, in order to talk about "the" degree, you need to first specify that very ample line bundle. $\endgroup$ Feb 10, 2014 at 23:07
  • $\begingroup$ @Starr: I have removed the question using the term "minimum degree". I hope the present question makes sense. $\endgroup$
    – user43198
    Feb 10, 2014 at 23:18
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    $\begingroup$ @user43198 Note that the number $C^2$ changes if you see the curve $C$ as embedded into another smooth surface. So, $C^2$ makes no sense for a curve in ${\mathbb P}^3$. (Well, it makes sense and is equal to $0$.) $\endgroup$ Feb 11, 2014 at 6:21

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