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Curve singularities with one Puiseux pair and value sets of modules over their local rings

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dc.contributor.authorAlberich-Carramiñana, María
dc.contributor.authorAlmirón, Patricio
dc.contributor.authorMoyano-Fernández, Julio-José
dc.date.accessioned2023-06-17T08:27:51Z
dc.date.available2023-06-17T08:27:51Z
dc.date.issued2021-05
dc.description.abstractIn this paper we characterize the value set ∆ of the R-modules of the form R+zR for the local ring R associated to a germ ξ of an irreducible plane curve singularity with one Puiseux pair. In the particular case of the module of Kähler differentials attached to ξ , we recover some results of Delorme. From our characterization of ∆ we introduce a proper subset of semimodules over the value semigroup of the ring R. Moreover, we provide a geometric algorithm to construct all possible semimodules in this subset for a given value semigroup.
dc.description.departmentDepto. de Álgebra, Geometría y Topología
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.facultyInstituto de Matemática Interdisciplinar (IMI)
dc.description.refereedFALSE
dc.description.sponsorshipMinisterio de Ciencia, Innovación y Universidades
dc.description.sponsorshipGeneralitat de Catalunya
dc.description.sponsorshipUniversitat Jaume I
dc.description.statusunpub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/72682
dc.identifier.urihttps://hdl.handle.net/20.500.14352/7198
dc.language.isoeng
dc.relation.projectIDMTM2015- 69135-P; MTM2016-76868- C2-1-P; PGC2018-096446-B-C22
dc.relation.projectID2017SGR-932; MDM-2014-0445
dc.relation.projectIDUJI-B2018-10
dc.rights.accessRightsopen access
dc.subject.cdu512.7
dc.subject.keywordR-modules
dc.subject.keywordΓ-semimodules
dc.subject.keywordCurve singularities
dc.subject.keywordModuli
dc.subject.keywordValue sets
dc.subject.ucmÁlgebra
dc.subject.ucmGeometria algebraica
dc.subject.ucmGrupos (Matemáticas)
dc.subject.unesco1201 Álgebra
dc.subject.unesco1201.01 Geometría Algebraica
dc.titleCurve singularities with one Puiseux pair and value sets of modules over their local rings
dc.typejournal article
dcterms.references1. P. Almirón, J.J. Moyano-Fernández, A formula for the conductor of a semimodule of a numerical semigroup with two generators, Semigroup Forum (2021), https://doi.org/10.1007/s00233-021-10182-1. 2. V.I. Arnol’d, Critical points of smooth functions, Proc. Int. Congress Math. Vancouver, Vol.1 (1974), 19–39. 3. I. Cherednik and I. Philipp, Modules over plane curve singularities in any ranks and DAHA, J. Algebra 520 (2019), 186–236. 4. C. Delorme, Sur les modules des singularités des courbes planes, Bull. Soc. Math. France 106 (1978), 417–446. 5. G.-M Greuel, H Knörrer Einfache Kurvensingularitäten und torsionfreie Moduln, Math. Ann, 270 (1985), 417-425. 6. G.-M Greuel, G Pfister Moduli spaces for torsion free modules on curve singularities, I, J. Algebraic Geom., 2 (1993), 81-135. 7. A. Hefez, M.E. Hernandes, Standard bases for local rings of branches and their modules of differentials, J. Symbolic Comput.,42 (2007), 178-191. 8. A. Hefez, M.E. Hernandes, The analytic classification of plane branches, Bull. Lond. Math. Soc.,43 (2011), 289-298. 9. J.J. Moyano-Fernández, J. Uliczka, Lattice paths with given number of turns and numerical semigroups, Sem. Forum 88, no. 3, (2014), 631–646. 10. J.J. Moyano-Fernández, J. Uliczka, Duality and syzygies for semimodules over numerical semigroups, Sem. Forum 92, no. 3, (2016), 675–690. 11. G. Pfister and J. H. M. Steenbrink, Reduced Hilbert schemes for irreducible curve singularities, J. Pure Appl. Algebra 77, no.1 (1992), 103–116. 12. J. Piontkowski, Topology of the compactified Jacobians of singular curves, Math. Z. 55, no.1 (2007), 195–226. 13. J. L. Ramírez Alfonsín, The Diophantine Frobenius problem, Lecture Series in Mathematics and Its Applications 30, Oxford U.P. (2005). 14. J. C. Rosales, P. A. García Sanchez, Numerical Semigroups, Springer (2009). 15. B. Teissier, Appendix, in [16], 1986. 16. O. Zariski, Le probleme des modules pour les branches planes, with an appendix by B. Teissier Hermann, 1986
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