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The Hilbert scheme of Buchsbaum space curves. (Le schéma de Hilbert des courbes gauches de Buchsbaum.) (English. French summary) Zbl 1271.14007
Let \(C\) be a curve in \(\mathbb P^3\) over an algebraically closed field \(k\) and let \(R = k[x_0,x_1,x_2,x_3]\). Denote by \(I(C)\) its homogeneous ideal, and by \(\mathcal I_C\) its ideal sheaf. The Rao module (sometimes Hartshorne-Rao module or deficiency module) of \(C\) is the graded \(R\)-module \(M = H_*^1(\mathcal I_C) := \bigoplus_{t \in \mathbb Z} H^1(\mathbb P^3, \mathcal I_C (t))\). The curve \(C\) is said to be Buchsbaum if \(M\) is annihilated by the irrelevant ideal of \(R\). This is true in particular when \(M\) has only one non-zero component, and such curves are the central object of study of this paper. Assume from now on that \(C\) is such a curve. We consider all the components, \(V\), of the Hilbert scheme \(H(d,g)\) that contain \(C\). The author determines \(V\) from the point of view of describing the graded Betti numbers of the generic curve of \(V\) in terms of the graded Betti numbers of \(C\). This deep and careful study of the behavior of the Betti numbers of curves in these families, combined with earlier work of the author, gives us a good understanding of the Hilbert scheme of \(C\), and its singular locus.

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14H50 Plane and space curves
14M06 Linkage
13D02 Syzygies, resolutions, complexes and commutative rings
13C40 Linkage, complete intersections and determinantal ideals
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