# ANNALES DE L'INSTITUT FOURIER

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

We study classical and new invariants of plane curve singularities $f\in K\left[\left[x,y\right]\right]$, $K$ an algebraically closed field of characteristic $p\ge 0$. It is known, in characteristic zero, that $\kappa \left(f\right)=2\delta \left(f\right)-r\left(f\right)+\mathrm{mt}\left(f\right)$, where $\kappa \left(f\right),\delta \left(f\right),r\left(f\right)$ and $\mathrm{mt}\left(f\right)$ denotes kappa invariant, delta invariant, number of branches and multiplicity of $f$ respctively. For arbitrary characteristic, by introducing new invariant $\gamma$, we prove in this note that $\kappa \left(f\right)\ge \gamma \left(f\right)+\mathrm{mt}\left(f\right)-1\ge 2\delta \left(f\right)-r\left(f\right)+\mathrm{mt}\left(f\right)$ 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 $C\subset {ℙ}^{2}$ (in fact, if $p>\kappa \left(f\right)$, resp. $p>degC\left(degC-1\right)$), then $f$, resp. $C$ has no wild vanishing cycle.

Nous étudions des invariants classiques et nouveaux des singularités de courbes planes $f\in K\left[\left[x,y\right]\right]$$K$ est un corps algébriquement clos de caractéristique $p\ge 0$. En caratéristique nulle, il est connu que $\kappa \left(f\right)=2\delta \left(f\right)-r\left(f\right)+\mathrm{mt}\left(f\right)$, où $\kappa \left(f\right),\delta \left(f\right),r\left(f\right)$ et $\mathrm{mt}\left(f\right)$ 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 $\gamma$, nous prouvons dans cette note que $\kappa \left(f\right)\ge \gamma \left(f\right)+\mathrm{mt}\left(f\right)-1\ge 2\delta \left(f\right)-r\left(f\right)+\mathrm{mt}\left(f\right)$ 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 $C\subset {ℙ}^{2}$ (c’est-à-dire, si $p>\kappa \left(f\right)$, resp. $p>degC\left(degC-1\right)$), alors $f$, resp. $C$ n’a pas de cycle évanescent sauvage.

Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3057
Classification: 14H20,  14B05
Keywords: Invariants of plane curve singularities, Plücker formulas, wild vanishing cycles
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.3057/

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