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We study classical invariants for plane curve singularities $f\in K[[x,y]]$,$K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number,delta invariant, kappa invariant and multiplicity. It is known, incharacteristic zero, that $\mu(f)=2\delta(f)-r(f)+1$ and that$\kappa(f)=2\delta(f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic,Deligne prove that there is always the inequality $\mu(f)\geq2\delta(f)-r(f)+1$ by showing that $\mu(f)-\left( 2\delta(f)-r(f)+1\right)$measures the wild vanishing cycles. By introducing new invariants$\gamma,\tilde{\gamma}$, we prove in this note that $\kappa(f)\geq\gamma(f)+\mathrm{mt}(f)-1\geq 2\delta(f)-r(f)+\mathrm{mt}(f)$ with equalitiesif and only if the characteristic $p$ does not divide the multiplicity of anybranch of $f$. As an application we show that if $p$ is "big" for $f$ (in fact$p > \kappa(f)$), then $f$ has no wild vanishing cycle. Moreover we obtain somePl\"ucker formulas for projective plane curves in positive characteristic.

Invariants of plane curve singularities and Pl\"ucker formulas in positive characteristic