# zbMATH — the first resource for mathematics

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.

##### MSC:
 32S30 Deformations of complex singularities; vanishing cycles 14H50 Plane and space curves 14B12 Local deformation theory, Artin approximation, etc.
##### Keywords:
free divisor; discriminant; space curves
Full Text:
##### References:
 [1] [B-G] Buchweitz, R.; Greuel, G.-M. The Milnor number and deformations of complex curve singularities. Inv. Math.58 (1980), 241–281 · Zbl 0458.32014 [2] [Buch] Buchweitz, R.: Contributions à la théorie des singularités. Thesis Paris VII 1981 [3] [Bur] Burch, L.: On ideals of finite homological dimension in local rings. Proc. Cambr. Phil. Soc.64 (1968), 941–948 · Zbl 0172.32302 [4] [B-C] Behnke, K.; Christophersen, J.: Hyperplane sections and obstructions (rational surface singularities). Comp. Math.77 (1991), 233–268 · Zbl 0728.14034 [5] [Da] J. Damon;Higher Multiplicities and Almost free Divisors and Complete Insersections, Preprint 1992 [6] [D-M] J. Damon and D. Mond;A-codimension and the vanishing topology of discriminants, Inv. Math.106, (1991), 217–242 · Zbl 0772.32023 [7] [Gr] Grauert, H.: Über die Deformationen isolierter Singularitäten analytischer Mengen. Inv. Math.15 (1972), 171–198 · Zbl 0237.32011 [8] [He] Herzog, J.: Deformationen von Cohen-Macaulay Algebren. J. f. Reine u. Angew. Math.318 (1980), 83–105 · Zbl 0425.13005 [9] [Hu] Huneke, C.: The koszul homology of an ideal. Adv. Math.56 (1985), 295–318 · Zbl 0585.13006 [10] [J-S1] de Jong, T.; van Straten, D.: On the base space of a semi-universal deformation of rational quadruple points. Ann. Math.134 (1991), 653–678 · Zbl 0765.32018 [11] [J-S2] de Jong, T.; van Straten, D.: Deformations of the normalization of hypersurfaces Math. Ann.288 (1990), 527–547 · Zbl 0715.32013 [12] [L] Looijenga, E: Isolated singular points on complete intersections. London Math. Soc. Lecture Notes Series77, Cambridge University Press, 1984 · Zbl 0552.14002 [13] [P] Palamodov, V.: Deformations of complex spaces. In: Several complex variables IV, Encyclopaedia of Math. Sciences; eds: Gindikin & Khenkin, Springer Verlag, Berlin & 1990 · Zbl 0721.32009 [14] [Sa] Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. 1A Math.27 (1980), 265–291 · Zbl 0496.32007 [15] [Sch] Schaps, M.: Deformations of Cohen-Macaulay schemes of codimension 2 and non-singular deformations of space curves. Am. J. Math.99 (1977), 669–684 · Zbl 0358.14006 [16] [S-V] Simis, A.; Vasconcelos, W.: The syzygies of the conormal module. Am. J. Math.103 (1980), 203–224 · Zbl 0467.13009 [17] [Tei] Teissier, B.: The hunting of invariants in the geometry of discriminants. In: Real and complex singularities. (Ed. by P. Holme) Sythoft & Noordhoff Int. Publ. Alphen a/d. Rijn, (1978) [18] [Ter] Terao, H.: The exponents of a free hypersurface. In: Singularities. Proc. Symp. Pure Math., Vol.40, Part 2, p. 561–566 [19] [Wah] Wahl, J.: Smoothings of normal surface singularities. Topology, Vol.20 (1981), 219–246 · Zbl 0484.14012 [20] [Wal] Waldi, R.: Deformationen von Gorenstein Singularitäten der Kodimension 3. Math. Ann.242 (1979), 201–208 · Zbl 0426.14004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.