We study classical invariants for plane curve singularities f∈K[[x,y]],
K an algebraically closed field of characteristic p≥0: Milnor number,
delta invariant, kappa invariant and multiplicity. It is known, in
characteristic zero, that μ(f)=2δ(f)−r(f)+1 and that
κ(f)=2δ(f)−r(f)+mt(f). For arbitrary characteristic,
Deligne prove that there is always the inequality μ(f)≥2δ(f)−r(f)+1 by showing that μ(f)−(2δ(f)−r(f)+1)
measures the wild vanishing cycles. By introducing new invariants
γ,γ~, we prove in this note that κ(f)≥γ(f)+mt(f)−1≥2δ(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 an application we show that if p is "big" for f (in fact
p>κ(f)), then f has no wild vanishing cycle. Moreover we obtain some
Pl\"ucker formulas for projective plane curves in positive characteristic.Comment: 15 pages; final version; to appear in the Annales de l'Institut
Fourie