The minimal Tjurina number of irreducible germs of plane curves singularities
| dc.contributor.author | Alberich-Carramiñana, María | |
| dc.contributor.author | Almirón, Patricio | |
| dc.contributor.author | Blanco, Guillem | |
| dc.contributor.author | Melle Hernández, Alejandro | |
| dc.date.accessioned | 2023-06-17T08:27:58Z | |
| dc.date.available | 2023-06-17T08:27:58Z | |
| dc.date.issued | 2020 | |
| dc.description.abstract | In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer [6], and by Wall and Mattei [13, 11]. | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.faculty | Instituto de Matemática Interdisciplinar (IMI) | |
| dc.description.refereed | FALSE | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovación (MICINN) | |
| dc.description.sponsorship | Generalitat de Catalunya | |
| dc.description.sponsorship | Barcelona Graduate School of Mathematics (BGSMath) | |
| dc.description.status | unpub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/72777 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/7204 | |
| dc.language.iso | eng | |
| dc.relation.projectID | MTM2015-69135-P; MTM2016-76868-C2-1-P | |
| dc.relation.projectID | 2017SGR-932 | |
| dc.relation.projectID | MDM-2014-0445 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 512.76/.77 | |
| dc.subject.keyword | Curve singularities | |
| dc.subject.keyword | Tjurina number | |
| dc.subject.keyword | Milnor number | |
| dc.subject.ucm | Funciones (Matemáticas) | |
| dc.subject.ucm | Geometria algebraica | |
| dc.subject.unesco | 1202 Análisis y Análisis Funcional | |
| dc.subject.unesco | 1201.01 Geometría Algebraica | |
| dc.title | The minimal Tjurina number of irreducible germs of plane curves singularities | |
| dc.type | journal article | |
| dcterms.references | [1] P. Almirón, G. Blanco, A note on a question of Dimca and Greuel, C. R. Math. Acad. Sci. Paris, Ser. I 357 (2019), 205–208. [2] J. Brian¸con, M. Granger, Ph. Maisonobe, Le nombre de modules du germe de courbe plane x a+y b = 0, Math. Ann. 279 (1988), 535–551. [3] E. Casas-Alvero, Singularities of plane curves, London Math. Soc. Lecture Note Ser. 276, Cambridge Univ. Press, Cambridge, 2000. [4] C. Delorme, Sur les modules des singularités des courbes planes, Bull. Soc. Math. France 106 (1978), 417–446. [5] A. Dimca, G.-M. Greuel, On 1-forms on isolated complete intersection on curve singularities, J. of Singul. 18 (2018), 114–118. [6] Y. Genzmer, Dimension of the moduli space of a curve in the complex plane, Preprint in: arXiv:1610.05998 (2016). [7] Y. Genzmer, M. E. Hernandes, On the Saito’s basis and the Tjurina Number for Plane Branches, Preprint in: arXiv:1904.03645 (2019). [8] G.-M. Greuel, C. Lossen, E. Shustin, Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Berlin, 2007. [9] L. D. Tráng, C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), no. 1, 67–78. [10] I. Luengo, G. Pfister, Normal forms and moduli spaces of curve singularities with semigroup h2p, 2q, 2pq + di, Compos. Math. 76 (1990), no. 1–2, 247–264. [11] J. F. Mattei, Modules de feuilletages holomorphes singuliers: I équisingularité, Invent. Math. 103 (1991), no. 2, 297–325. [12] R. Peraire, Tjurina number of a generic irreducible curve singularity, J. Algebra 196 (1997), no.1,114–157. [13] C. T. C. Wall, Notes on the classification of singularities, Proc. London Math. Soc. 48 (1984), no.3,461–513. [14] C. T. C. Wall, Singular points of plane curves, London Math. Soc. Students Texts 63, Cambridge Univ.Press, Cambridge, 2004. [15] B. Teissier, Appendix, in [16], 1986. [16] O. Zariski, Le probléme des modules pour les branches planes, Hermann, Paris, 1986. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | c5f952f6-669f-4e3d-abc8-76d6ac56119b | |
| relation.isAuthorOfPublication.latestForDiscovery | c5f952f6-669f-4e3d-abc8-76d6ac56119b |
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