Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T08:26:49.109Z Has data issue: false hasContentIssue false
  • English
  • Français

The log-canonical threshold of a plane curve

Published online by Cambridge University Press:  11 February 2016

CARLOS GALINDO ,
FERNANDO HERNANDO  and
FRANCISCO MONSERRAT
CARLOS GALINDO
Affiliation:
Instituto Universitario de Matemáticas y Aplicaciones de Castellón and Departamento de Matemáticas, Universitat Jaume I, Campus de Riu Sec. 12071 Castelló, Spain. e-mail: [email protected]; [email protected]
FERNANDO HERNANDO
Affiliation:
Instituto Universitario de Matemáticas y Aplicaciones de Castellón and Departamento de Matemáticas, Universitat Jaume I, Campus de Riu Sec. 12071 Castelló, Spain. e-mail: [email protected]; [email protected]
FRANCISCO MONSERRAT
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in [27].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 3 , May 2016 , pp. 513 - 535
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alberich–Carramiñana, M., Alvarez-Montaner, J. and Dachs–Cadefau, F.. Multiplier ideals in two-dimensional local rings with rational singularities. ArXiv:1412.3605.Google Scholar
[2]Alberich–Carramiñana, M., Alvarez–Montaner, J., Dachs–Cadefau, F. and González–Alonso, V.. Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities. ArXiv:1412.3607.Google Scholar
[3]Aprodu, M. and Naie, D.Enriques diagrams and log-canonical thresholds of curves on smooth surfaces. Geom. Dedicata 146 (2010), 43–66.CrossRefGoogle Scholar
[4]Artal Bartolo, E., Cassou–Nogués, P., Luengo, I. and Melle–Hernández, A.. On the log-canonical threshold for germs of plane curves. Contemp. Math. 474 (2008), 1–14.CrossRefGoogle Scholar
[5]Atiyah, M.F.Resolution of singularities and division of distributions. Comm. Pure Appl. Math. 23 (1970), 145–150.CrossRefGoogle Scholar
[6]Birkar, C.Ascending chain condition for log-canonical thresholds and termination of log flips. Duke Math. J. 136 (2007), 173–180.CrossRefGoogle Scholar
[7]Birkar, C., Cascini, P., Hacon, C. and McKernan, J.. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), 405–468.CrossRefGoogle Scholar
[8]Brieskorn, E. and Knorrer, H.Plane Algebraic Curves (Birkhäuser, Basel, 1986).CrossRefGoogle Scholar
[9]Budur, N.On Hodge spectrum and multiplier ideals. Math. Ann. 327 (2003), 257–270.CrossRefGoogle Scholar
[10]Budur, N., González–Pérez, P.D. and González–Villa, M.. Log-canonical thresholds of quasi-ordinary hypersurface singularities. Proc. Amer. Math. Soc. 140 (2012), 4075–4083.CrossRefGoogle Scholar
[11]Budur, N., Mustaţă, M. and Saito, M.. Bernstein-Sato polynomials of arbitrary varieties. Compositio Math. 142 (2006), 779–797.CrossRefGoogle Scholar
[12]Campillo, A.Algebroid curves in positive characteristic. Lecture Notes in Mathematics vol. 613, (Springer-Verlag, 1980).CrossRefGoogle Scholar
[13]Campillo, A., Delgado, F. and Gusein–Zade, S.M.On generators of the semigroup of a plane curve singularity. J. London Math. Soc. 60 (1999), 420–430.CrossRefGoogle Scholar
[14]Campillo, A., González–Sprinberg, G. and Lejeune–Jalabert, M.. Clusters of infinitely near points. Math. Ann. 306 (1996), 169–194.CrossRefGoogle Scholar
[15]Delgado, F.The semigroup of values of a curve singularity with several branches. Manuscripta Math. 59 (1987), 347–374.Google Scholar
[16]Delgado, F. An arithmetical factorisation for the critical point set of some maps from ${\mathbb{C}}^2$ to ${\mathbb{C}}^2$. Singularities - Lille 1991. London Mathematical Society Lecture Note Series 201 (ed. Brasselet, Jean-Paul) (Cambridge University Press, 1994), 61–100.Google Scholar
[17]Ein, L., Lazarsfeld, R., Smith, K.E. and Varolin, D.. Jumping coefficients of multiplier ideals. Duke Math. J. 123 (2004), 469–506.CrossRefGoogle Scholar
[18]Favre, C. and Jonsson, M.Valuations and multiplier ideals. J. Amer. Math. Soc. 18 (2005), 655–684.CrossRefGoogle Scholar
[19]Galindo, C. and Monserrat, F.The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface. Adv. Math. 225 (2010), 1046–1068.CrossRefGoogle Scholar
[20]Halle, L.H. and Nicaise, J.Motivic zeta functions of abelian varieties and the monodromy conjecture. Adv. Math. 227 (2011), 610–653.CrossRefGoogle Scholar
[21]Hara, N. and Yoshida, K.I.A generalization of tight closure and multiplier ideals. Trans. Amer. Math. Soc. 355 (2003), 3143–3174.CrossRefGoogle Scholar
[22]Hyry, E. and Järvilehto, T.. Jumping numbers and ordered tree structures on the dual graph. Manuscripta Math. 136 (2011), 411–437.CrossRefGoogle Scholar
[23]Igusa, J.Critical points of smooth functions and their normal forms. Complex and Algebraic Geometry. (Iwanami Shoten, 1977), 357–368.CrossRefGoogle Scholar
[24]Järvilehto, T.. Jumping numbers of a simple complete ideal in a two-dimensional regular local ring. Mem. Amer. Math. Soc. 214 (2011), no. 1009.Google Scholar
[25]Kollár, J.Singularities of pairs. Proc. Sympos. Pure Math. 62 (1997), 221–287.CrossRefGoogle Scholar
[26]Kollár, J.et al.Flips and abundance for algebraic threefolds. Astérisque 211 (1992).Google Scholar
[27]Kuwata, T.On log canonical thresholds of reducible curves. Amer. J. Math. 121 (1999), 701–721.CrossRefGoogle Scholar
[28]Lazarsfeld, R.Positivity in Algebraic Geometry. Vol II (Springer, 2004).Google Scholar
[29]Lipman, J.Adjoints of ideals in regular local rings rings. Math. Res. Lett. 1 (1994), 739–755, with an appendix by S.D. Cutkosky.CrossRefGoogle Scholar
[30]Lipman, J. and Sathaye, A.Jacobian ideals and theorem of Briançon–Skoda about integral closures of ideals. Michigan Math. J. 28 (1981), 199–222.Google Scholar
[31]Mustaţă, M.Singularities of pairs via jet schemes. J. Amer. Math. Soc. 15 (2002), 599–615.CrossRefGoogle Scholar
[32]Naie, D.Jumping numbers of a unibranch curve on a smooth surface. Manuscripta Math. 128 (2009), 33–49.CrossRefGoogle Scholar
[33]Shokurov, V. V.Three-dimensional log perestroikas (in Russian). With an appendix in English by Kawamata, Y.. Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 105-203, translation in Russian Acad. Sci. Izv. Math. 40 (1993), 95–202.Google Scholar
[34]Smith, K.E. and Thompson, H.M.Irrelevant exceptional divisors for curves on a smooth surface. Contemp. Math. 448 (2007), 245–254.CrossRefGoogle Scholar
[35]Steenbrink, J.H.M.The spectrum of hypersurface singularities. Astérisque 179–180 (1989), 163–184.Google Scholar
[36]Tucker, K.Jumping numbers on algebraic surfaces with rational singularities. Trans. Amer. Math. Soc. 362 (2010), 3223–3241.CrossRefGoogle Scholar
[37]Tucker, K. Jumping numbers and multiplier ideals on algebraic surfaces. Ph. D. thesis. University of Michigan (2010).Google Scholar
[38]Varchenko, A.N.Asymptotic Hodge structures in the vanishing cohomology. Math. USSR Izv. 18 (1982), 469–512.CrossRefGoogle Scholar
[39]Varchenko, A.N.The complex singularity index does not change along the stratum μ=const (in Russian). Funktsional. Anal. i Prilozhen 16 (1982), 1–12.CrossRefGoogle Scholar
[40]Varchenko, A.N.Semicontinuity of the complex singularity exponent (in Russian). Funksional. Anal. i Prilozhen 17 (1983), 77–78.Google Scholar
[41]Zariski, O. and Samuel, P.Commutative Algebra. Vol II (Springer-Verlag, 1960).CrossRefGoogle Scholar