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Dec
2
comment Global sections of the structure sheaf of a non-reduced projective scheme
@Litt: To answer your question, we can simply compute the arithmetic genus of the curve mentioned above using the adjunction formula and other standard tools. It is going to be negative. As far as I remember arithmetic genus of a complete intersection curve must be non-negative. Also, in a more basic way , a curve of small degree cannot be a complete intersection curve in a surface of large degree. Another way of seeing this is the description of the ideal of the curve as given in the reference mentioned below in my comment to abx.
Dec
2
comment Global sections of the structure sheaf of a non-reduced projective scheme
@Litt: I agree with your argument in the case $C$ is a complete intersection. But a very general surface will not contain $C$ (as in the example). Even more, any surface containing $C$ will not have Pic$(X)=\mathbb{Z}$. What is your argument to prove that a very general degree $d$ surface contains $C$?
Dec
2
comment Global sections of the structure sheaf of a non-reduced projective scheme
@Litt: First, $\mathcal{O}_X(C)$ is a locally free $\mathcal{O}_X$-module. So, tensor product by $K_X$, not $K_C$. Second, I am interested in a problem slightly different from what you answer. I would like to know, given a curve (especially my example) what is the dimension of the space of global sections. I do not totally understand the statement you are proving. The curve that I mentioned above will almost never be a complete intersection. May be you could elaborate a little bit.
Dec
2
revised Global sections of the structure sheaf of a non-reduced projective scheme
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Dec
2
revised Global sections of the structure sheaf of a non-reduced projective scheme
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Dec
2
revised Global sections of the structure sheaf of a non-reduced projective scheme
edited title
Dec
2
comment Global sections of the structure sheaf of a non-reduced projective scheme
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.
Dec
2
comment Global sections of the structure sheaf of a non-reduced projective scheme
Why the down vote? Is it a trivial question or is there something very unclear?
Dec
2
comment Global sections of the structure sheaf of a non-reduced projective scheme
$C$ 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.
Dec
2
revised Global sections of the structure sheaf of a non-reduced projective scheme
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Dec
2
asked Global sections of the structure sheaf of a non-reduced projective scheme
Nov
29
comment Hypersurfaces with Picard group generated by classes of lines on the same plane
@Huizenga: The motivation comes from the study of Noether-Lefschetz locus which I unfortunately do not think is possible to explain in a paragraph. I am sorry. I would expect that for $d$ large enough this phenomenon happens. Again this is motivated by results on Noether-Lefschetz locus.
Nov
29
comment Hypersurfaces with Picard group generated by classes of lines on the same plane
@Bright: The base field in $\mathbb{C}$.
Nov
28
asked Hypersurfaces with Picard group generated by classes of lines on the same plane
Nov
23
revised Deformation of a family of curves in a surface
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Nov
23
comment Deformation of a family of curves in a surface
@Huizenga: Sorry, I meant open subset of $B$.
Nov
23
revised Deformation of a family of curves in a surface
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Nov
23
comment Deformation of a family of curves in a surface
@quim: The language in the question has been totally modified.
Nov
23
revised Deformation of a family of curves in a surface
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Nov
22
accepted Does flatness/smoothness over special fiber imply flatness/smoothness globally?