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Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.

What is the expected number $N$ of conditions imposed by requiring the surface to have multiplicity $\beta$ along the curve?

If $\binom{k+3}{3}-N>0$ is it true that for $k\gg 0$ the general element of $\mathcal{S}$ is an irreducible surface having exactly multiplicity $\beta$ along $C$ and smooth outside $C$?

If the answer at the first question if yes and $S$ is a general surface in $\mathcal{S}$ is it true that blowing-up $\mathbb{P}^3$ along $C$ and taking the strict transform of $S$ we get a smooth divisor in the blow-up ?

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Let $\mathcal{I}_C$ be the sheaf of ideals defining $C$ in $\mathbb{P}^3$. You are asking for the expected codimension $N_k$ of $H^0(\mathbb{P}^3, \mathcal{I}_C^\beta(k))$ in $H^0(\mathbb{P}^3, \mathcal{O}_{\mathbb{P}^3}(k))$. Since $H^i(\mathbb{P}^3, \mathcal{I}_C^\beta(k))=0$ for $i>0$ and $k>>0$, we have $N_k=h^0( \mathcal{O}_{\mathbb{P}}/\mathcal{I}_C^\beta(k))=\chi (\mathcal{O}_{\mathbb{P}}/\mathcal{I}_C^\beta(k)))$ for $k>>0$.

Because of the exact sequences $0\rightarrow \mathcal{I}_C^{m-1}/\mathcal{I}_C^m\rightarrow \mathcal{O}_{\mathbb{P}}/\mathcal{I}_C^m\rightarrow \mathcal{O}_{\mathbb{P}}/\mathcal{I}_C^{m-1}\rightarrow 0$ and the isomorphisms $\mathcal{I}_C^{m-1}/\mathcal{I}_C^m\cong \mathrm{Sym}^{m-1}(\mathcal{I}_C/\mathcal{I}_C^2)$ we have $N_k= \chi (\mathcal{O}_C(k))+\chi (\mathcal{I}_C/\mathcal{I}_C^2(k))+\ldots+\chi (\mathrm{Sym}^{\beta-1}(\mathcal{I}_C/\mathcal{I}_C^2(k)))$. We have $\deg(\mathcal{I}_C/\mathcal{I}_C^2)=\deg(\Omega ^1_{\mathbb{P}^3}\,|C) -\deg(\Omega ^1_C)=-4d+2-2g$ . Using Riemann-Roch one finds (if I didn't make mistakes) $$N_k=\binom{\beta+1} {2}(kd+1-g)-2\binom{\beta+1} {3}(2d+g-1)\ .$$

As for the other questions, let $b:P\rightarrow \mathbb{P}^3$ be the blowing up of $C$ in $\mathbb{P^3}$, $E$ the exceptional divisor, $H$ a hyperplane in $\mathbb{P}^3$. If $\mathcal{I}_C$ is generated by forms of degree $c$, then the linear system $|cb^*H-E|$ on $P$ is base point free, so $kb^*H-\beta E$ is base point free for $k\geq \beta c$. Moreover for $k>>0$ it is ample. If both conditions hold, the system contains a smooth irreducible surface by the Bertini theorem.

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  • $\begingroup$ Thank you for the answer. I just do not understand where you take into account the multiplicity $\beta$ along $C$. It seems to me that in this way you compute the dimension of the system of surfaces having multiplicity $2$ along $C$. Am I missing something? $\endgroup$
    – user47036
    Commented Apr 6, 2014 at 17:24
  • $\begingroup$ No, sorry, I just did it for $\beta=2$. The same method works for any $\beta$ but the computation becomes tedious. The rest of the argument adapts immediately. $\endgroup$
    – abx
    Commented Apr 6, 2014 at 19:17
  • $\begingroup$ I have edited the question and treated the general case. $\endgroup$
    – abx
    Commented Apr 7, 2014 at 5:11
  • $\begingroup$ Can you say a quick word about how you compute the Euler characteristics of the symmetric powers that pop up? $\endgroup$
    – Lalit Jain
    Commented Apr 7, 2014 at 16:35
  • $\begingroup$ You just need the degree (by Riemann-Roch), that is $c_1$, and that's standard: if $E$ has degree $d$ and rank 2, $\mathrm{Sym}^pE$ has degree $\frac{1}{2}dp(p+1) $ (by the splitting principle you just need to check that when $E=L\oplus M$, then it is straightforward). $\endgroup$
    – abx
    Commented Apr 7, 2014 at 19:27

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