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A note on the discriminant of a space curve. (English) Zbl 0858.32031
Let \(X\to B\) be a deformation of a germ of an analytic space \(X_0\) over a smooth base germ. In many situations the discriminant \(\Delta\) is a hypersurface with remarkable properties, like being a free divisor (meaning that the module of vector fields on \(B\), logarithmic along \(\Delta\), is free).
The author formulates a criterion for freeness of the discriminant in terms of relative \(T^1\)s. The author applies his criterion to space curves. The proof comes about from the combination of two facts. For families \(X\to S\) of curves with smooth general fibre the essential ingredient is that \(\omega^*_{X/S}\), the dual of the relative dualising sheaf, is flat over \(S\). On the other hand, this ‘\(\omega^*\)-constant’ property holds quite general for codimension two Cohen-Macaulay germs.

32S30 Deformations of complex singularities; vanishing cycles
14H50 Plane and space curves
14B12 Local deformation theory, Artin approximation, etc.
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