Invariants of plane curve singularities and Plücker formulas in positive characteristic

Abstract

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 $\kappa (f)=2\delta (f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $\mu (f)\ge 2\delta (f)-r(f)+1$ 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 $\kappa (f)\ge \gamma (f)+\mathrm{mt}(f)-1\ge 2\delta (f)-r(f)+\mathrm{mt}(f)$ 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 Plücker formulas for projective plane curves in positive characteristic.

arXiv comment15 pages; final version; to appear in the Annales de l'Institut Fourier