A note on the plane curve singularities in positive characteristic

TL;DR

This paper investigates plane curve singularities over fields of positive characteristic, introducing a modified Milnor number that remains invariant and establishing inequalities relating it to Newton polygons, with equality in the non-degenerate case.

Contribution

It provides a simple proof of an inequality relating the modified Milnor number and Newton polygon invariants, and shows equality holds for non-degenerate curves.

Findings

Proves the inequality ar  - ()  r() - r()  0.

Establishes ar  = () when the curve is non-degenerate.

Introduces explicit formulas for Newton polygon invariants.

Abstract

Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\geq 0$ we consider the Milnor number $\mu(f)$, the delta invariant $\delta(f)$ and the number $r(f)$ of its irreducible components. Put $\bar \mu(f)=2\delta(f)-r(f)+1$. If $p=0$ then $\bar \mu (f)=\mu(f)$ (the Milnor formula). If $p>0$ then $\mu(f)$ is not an invariant and $\bar \mu(f)$ plays the role of $\mu(f)$. Let $\mathcal N_f$ be the Newton polygon of $f$. We define the numbers $\mu(\mathcal N_{f})$ 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 $\bar \mu(f)-\mu(\mathcal N_{f})\geq r(\mathcal N_{f})- r(f)\geq 0$ due to Boubakri, Greuel and Markwig. We also prove that $\bar \mu(f)=\mu(\mathcal N_{f})$ when $f$ is non-degenerate.