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Milnor number equals Tjurina number for functions on space curves

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Mond, D. (David) and Straten, Duco van. (2001) Milnor number equals Tjurina number for functions on space curves. Journal of the London Mathematical Society , Vol.63 (No.1). pp. 177-187. ISSN 0024-6107

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Official URL: http://dx.doi.org/10.1112/S0024610700001320

Abstract

The equality of the Milnor number and Tjurina number for functions on space curve singularities, as conjectured recently by V. Goryunov, is proved. As a consequence, the discriminant in such a situation is a free divisor.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Singularities (Mathematics), Cohomology operations, Algebraic topology, Deformations of singularities, Curves, Algebraic
Journal or Publication Title: Journal of the London Mathematical Society
Publisher: Cambridge University Press
ISSN: 0024-6107
Date: February 2001
Volume: Vol.63
Number: No.1
Page Range: pp. 177-187
Identification Number: 10.1112/S0024610700001320
Status: Peer Reviewed
Access rights to Published version: Open Access
References: 1. B. Angéniol and M. Lejeune-Jalabert, Calcul differentiel et classes characteristique en geometrie algèbrique, Travaux en Cours 38 (Hermann, Paris, 1989). 2. L. Avramov and J. Herzog, `The Koszul algebra of a codimension two embedding', Math. Z. 175 (1980) 249±260. 3. K. Behnke and J. A. Christophersen, `Hypersurface sections and obstructions (rational surface singularities), Compositio Math. 77 (1991) 233±268. 4. R. Berger, Über verschiedene Differentenbegriffe, Berichte Heidelberger Akademie der Wissenschaften I. Abhandlungen (1960). 5. J. W. Bruce and R. M. Roberts, ` Critical points of functions on analytic varieties ', Topology 27 (1988) 57±90. 6. W. Bruns and J. Herzog, Cohen±Macaulay rings, Cambridge Studies in Advanced Mathematics 39 (Cambridge University Press, 1993). 7. D. A. Buchsbaum and D. S. Rim, `A generalised Koszul complex II ', Trans. AMS 111 (1964) 197±224. 8. R. Buchweitz, On Zariski's criterion for equisingularity and non-smoothable monomial curves, thesis, Universite! Paris VII, 1981. 9. V. Goryunov, `Functions on space curves', J. London Math. Soc. (2) 61 (2000) 807±822. 10. G.-M. Greuel and R.-O. Buchweitz, `The Milnor number and deformations of complex curve singularities ', Invent. Math. 58 (1980) 241±281. 11. G.-M. Greuel and E. Looijenga, `The dimension of smoothing components', Duke Math. J. 52 (1985) 263±272. 12. J. Herzog, `Deformationen von Cohen±Macaulay Algebren', J. Reine Angew. Math. 318 (1980) 83±105. 13. J. Herzog and R. Waldi, `Cotangent functors of curve singularities ', Manuscripta Math. 55 (1986) 307±341. 14. C. Huneke, `Linkage and the Koszul homology of ideals ', Amer. J. Math. 104 (1982) 1043±1062. 15. J. Montaldi and D. van Straten, `One-forms on singular curves and the topology of real curve singularities ', Topology 29 (1990) 501±510. 16. K. Saito, `Quasihomogene isolierte Singularitäten von Hyperächen', Inv. Math. 14 (1971) 123±142. 17. M. Schaps, `Deformations of Cohen±Macaulay schemes of codimension 2 and non-singular deformations of space-curves', Amer. J. Math. 99 (1977) 660±684. 18. D. van Straten, `A note on the discriminant of a space curve', Manuscripta Math. 87 (1995) 167±177. 19. R. Waldi, `Deformationen von Gorenstein Singularita$ten der Kodimension', Math. Ann. 242 (1979) 201±208.
URI: http://wrap.warwick.ac.uk/id/eprint/810

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