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Maximal rank curves and singular points of the Hilbert scheme. (English) Zbl 0724.14018
A curve X in \({\mathbb{P}}^ 3\) is said to be obstructed if the corresponding point of the Hilbert scheme is singular. A long-standing problem has been to give a geometrical characterization of non- obstructedness. Let \({\mathcal I}_ X\) be the ideal sheaf of X in \({\mathbb{P}}^ 3\). A curve is said to be of maximal rank if for each n, at most one of \(h^ 0({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) and \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is non-zero. A curve is said to have natural cohomology if at most one of \(h^ 0({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\), \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) and \(h^ 2({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is non-zero for each n. Prior to this paper, several examples were known of smooth obstructed space curves, but it was an open question (due to E. Sernesi) whether there exists one which furthermore has maximal rank.
For much of the paper X is assumed to have the property that \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is nonzero for exactly one n. The first set of results go to show that the “generic” such X (in a precise sense) is non-obstructed. The second part of the paper gives some useful techniques for constructing obstructed curves, using liaison. The authors apply these techniques in the third part to produce a concrete example of a smooth, obstructed curve with maximal rank.
The authors point out that their examples of a smooth obstructed maximal rank curve has also been constructed independently (and with different techniques) by C. Walter (“Some examples of obstructed curves in \({\mathbb{P}}^ 3\)”, Proc., Trieste 1989). They also point out that their example does not have natural cohomology, and indeed Walter asked whether all curves with this property may be non-obstructed. However, recently M. Martin-Deschamps and D. Perrin have produced a curve (but not a smooth one) which has natural cohomology but is obstructed (“Courbes gauches et modules de Rao”).

MSC:
14H50 Plane and space curves
14M06 Linkage
14C05 Parametrization (Chow and Hilbert schemes)
Software:
Macaulay2
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References:
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