论文标题
平面曲线奇异性的注释
A note on the plane curve singularities in positive characteristic
论文作者
论文摘要
给定代数曲线$ f = 0 $在代数封闭的特征$ p \ geq 0 $上，我们考虑milnor数字$μ（f）$，delta不变性$δ（f）$和数量$ r（f）$ r（f）$ r（f）$ r（f）$ r（f）$ r（f）$ r（f）$ r（f）。放置$ \barμ（f）=2δ（f）-r（f）+1 $。如果$ p = 0 $，则$ \barμ（f）=μ（f）$（milnor公式）。如果$ p> 0 $，则$μ（f）$不是不变的，$ \barμ（f）$扮演$μ（f）$。令$ \ Mathcal n_f $为$ f $的牛顿多边形。我们定义数字$μ（\ Mathcal n_ {f}）$和$ r（\ Mathcal n_ {f}）$，可以通过显式公式计算。本说明的目的是简单地证明不等式$ \barμ（f）-μ（\ Mathcal n_ {f}）\ geq r（\ Mathcal n_ {f}） - r（f）\ geq 0 $归因于Boubakri，Greuel，Greuel，Greuel和Markwig。我们还证明$ \barμ（f）=μ（\ Mathcal n_ {f}）$当$ f $是非分级时。
Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\geq 0$ we consider the Milnor number $μ(f)$, the delta invariant $δ(f)$ and the number $r(f)$ of its irreducible components. Put $\bar μ(f)=2δ(f)-r(f)+1$. If $p=0$ then $\bar μ(f)=μ(f)$ (the Milnor formula). If $p>0$ then $μ(f)$ is not an invariant and $\bar μ(f)$ 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 $\bar μ(f)-μ(\mathcal N_{f})\geq r(\mathcal N_{f})- r(f)\geq 0$ due to Boubakri, Greuel and Markwig. We also prove that $\bar μ(f)=μ(\mathcal N_{f})$ when $f$ is non-degenerate.
