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Une remarque sur la lissification des courbes gauches. (A remark about smoothability of space curves). (French) Zbl 0752.14030

One way of constructing smooth space curves with “good” properties is by smoothing reducible nodal curves. The usual smoothing criterion is the following one:
A nodal curve \(X\) in \(\mathbb{P}^ 3\) is smoothable in case: (i) it defines a smooth point of the Hilbert scheme; and (ii) for each node \(x\) of \(X\), there is a first order infinitesimal deformation of \(X\) inside \(\mathbb{P}^ 3\) inducing a non-trivial infinitesimal deformation of the germ \((X,x)\).
In this note, the authors produce examples of smoothable nodal space curves, satisfying the first condition of the above criterion, but such that every first order infinitesimal deformation of the curve inside \(\mathbb{P}^ 3\) induces the trivial infinitesimal deformation of each of its nodes.

MSC:

14H50 Plane and space curves
14H20 Singularities of curves, local rings
14D15 Formal methods and deformations in algebraic geometry
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