Invariants of plane curve singularities and Plücker formulas in positive characteristic
Annales de l'Institut Fourier, Volume 66 (2016) no. 5, p. 2047-2066

We study classical and new invariants of plane curve singularities fK[[x,y]], K an algebraically closed field of characteristic p0. It is known, in characteristic zero, that κ(f)=2δ(f)-r(f)+mt(f), where κ(f),δ(f),r(f) and mt(f) denotes kappa invariant, delta invariant, number of branches and multiplicity of f respctively. For arbitrary characteristic, by introducing new invariant γ, we prove in this note that κ(f)γ(f)+mt(f)-12δ(f)-r(f)+mt(f) with equalities if and only if the characteristic p does not divide the multiplicity of any branch of f. As applications we obtain some Plücker formulas for projective plane curves in positive characteristic. Moreover we show that if p is “big” for f, resp. for irreducble curve C 2 (in fact, if p>κ(f), resp. p>degC(degC-1)), then f, resp. C has no wild vanishing cycle.

Nous étudions des invariants classiques et nouveaux des singularités de courbes planes fK[[x,y]]K est un corps algébriquement clos de caractéristique p0. En caratéristique nulle, il est connu que κ(f)=2δ(f)-r(f)+mt(f), où κ(f),δ(f),r(f) et mt(f) respectivement, désignent l’invariant kappa, l’invariant delta, le nombre de branches et la multiplicité de f. En caractéristique arbitraire, en introduisant un nouvel invariant γ, nous prouvons dans cette note que κ(f)γ(f)+mt(f)-12δ(f)-r(f)+mt(f) avec égalités si et seulement si la caractéristique p ne divise pas la multiplicité de chaque branche de f. Comme applications, nous obtenons des formules de Plücker pour les courbes projectives planes en caractéristique positive. Nous montrons de plus, que si la caractéristique p est “grande” par rapport à f, respectivement par rapport à une courbe irréductible C 2 (c’est-à-dire, si p>κ(f), resp. p>degC(degC-1)), alors f, resp. C n’a pas de cycle évanescent sauvage.

Received : 2015-05-08
Revised : 2016-01-15
Accepted : 2016-02-18
Published online : 2016-07-28
DOI : https://doi.org/10.5802/aif.3057
Classification:  14H20,  14B05
Keywords: Invariants of plane curve singularities, Plücker formulas, wild vanishing cycles
@article{AIF_2016__66_5_2047_0,
     author = {Nguyen, Hong Duc},
     title = {Invariants of plane curve singularities and Pl\"ucker formulas in positive characteristic},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     pages = {2047-2066},
     doi = {10.5802/aif.3057},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_5_2047_0}
}
Nguyen, Hong Duc. Invariants of plane curve singularities and Plücker formulas in positive characteristic. Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 2047-2066. doi : 10.5802/aif.3057. http://www.numdam.org/item/AIF_2016__66_5_2047_0/

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