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Let $C$ be a curve in $\mathbb{P}^3$ which is of local complete intersection is some smooth surface in $\mathbb{P}^3$. Assume $C$ is non-reduced. What can we say about $h^0(\mathcal{O}_C)$? When can we say that $h^0(\mathcal{O}_C) \le 1$? The curve that I have is mind is of the form $2l+C'$ (seen as a divisor in a smooth surface in $\mathbb{P}^3$) where $l$ is a line and $C'$ is a reduced plane curve lying on the same plane as $l$. Any idea/reference for the direction of approaching this problem will be most appreciated.

EDIT Note that $h^0(\mathcal{O}_C)) \ge 1$ and is equal to $1$ if and only if $h^1(\mathcal{O}_X(-C))=h^1(\mathcal{O}_X(C)(d-4))$ vanish.

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    $\begingroup$ Why the down vote? Is it a trivial question or is there something very unclear? $\endgroup$
    – Naga Venkata
    Commented Dec 2, 2013 at 8:38

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Your question is probably not well formulated. What I understand is that the curve $C$ is actually a plane curve of the form $2l+C'$, with $l$ and $C'$ in $\mathbb{P}^2$. Then of course $h^0(\mathcal{O}_C)=1$, because of the exact sequence $\ 0\rightarrow \mathcal{O}_{\mathbb{P}^2}(-C)\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{O}_C\rightarrow 0 $.

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  • $\begingroup$ $C$ is not necessarily a plane curve. I do not know how else to write it. I meant that the curve is the scheme associated to an effective divisor on a surface. Using the adjunction formula one sees that the arithmetic genus depends on the degree of the surface. For example take a smooth surface containing $l, C'$. This will also contain the curve $2l+C'$ as a Weil divisor. But the genus depends on the degree of the surface. $\endgroup$
    – Naga Venkata
    Commented Dec 2, 2013 at 8:37
  • $\begingroup$ Sorry I don't get it. You say that "$C'$ is a reduced plane curve lying on the same plane as $l$". Doesn't that mean that $C$ is the curve $2l+C'$ in that plane? $\endgroup$
    – abx
    Commented Dec 2, 2013 at 8:51
  • $\begingroup$ No. The ideal of $2l+C'$ does not contain the ideal of the plane. See "Le Schema de Hilbert des Courbes gauches localement Cohen-Macualay n'est (presque) jamais reduit" by M Martin-Descamps and D. Perrin Proposition $0.6$ for such examples. $\endgroup$
    – Naga Venkata
    Commented Dec 2, 2013 at 9:59

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