A note on the plane curve singularities in positive characteristic

2022/07/29 by Barroso, Evelia R. García, Płoski, Arkadiusz
#14H20 #32S05 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.14523

Abstract

Given an algebroid plane curve f=0 over an algebraically closed field of characteristic p≥ 0 we consider the Milnor number μ(f), the delta invariant δ(f) and the number r(f) of its irreducible components. Put μ(f)=2δ(f)-r(f)+1. If p=0 then μ(f)=μ(f) (the Milnor formula). If p>0 then μ(f) is not an invariant and μ(f) plays the role of μ(f). Let \mathcal N_f be the Newton polygon of f. We define the numbers μ(\mathcal N_f) and r(\mathcal N_f) which can be computed by explicit formulas. The aim of this note is to give a simple proof of the inequality μ(f)-μ(\mathcal N_f)≥ r(\mathcal N_f)- r(f)≥ 0 due to Boubakri, Greuel and Markwig. We also prove that μ(f)=μ(\mathcal N_f) when f is non-degenerate.

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