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

We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\ge 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $\mu (f)=2\delta (f)-r(f)+1$ and that $\kappa (f)=2\delta (f)-r(f)+\mathrm{m}\mathrm{t}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $\mu(f)\geq 2\delta(f)-r(f)+1$byshowingthat$\mu(f)-\left( 2\delta(f)-r(f)+1\right)$measuresthewildvanishingcycles\mathrm{.}Byintroducingnewinvariants$\gamma,\tilde{\gamma}$,weproveinthisnotethat$\kappa(f)\geq \gamma(f)+$\mathrm{m}\mathrm{t}$(f)-1\geq 2\delta(f)-r(f)+$\mathrm{m}\mathrm{t}$(f)$ with equalities if and only if the characteristic $p$ does not divide the multiplicity of any branch 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 some Pl"ucker formulas for projective plane curves in positive characteristic.