On the Milnor Formula in Arbitrary Characteristic

Abstract

The Milnor formula μ = 2δ − r + 1 relates the Milnor number μ, the double point number δ and the number r of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality μ ≥ 2δ − r + 1 in arbitrary characteristic and showed that the equality μ = 2δ − r + 1 characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic p. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. 25:61–85, 2010) or if p is greater than the intersection number of the singularity with its generic polar (Nguyen Annales de l’Institut Fourier, Tome 66(5):2047–2066, 2016). Then we improve our result on the Milnor number of irreducible singularities (Bull. Lond. Math. Soc. 48:94–98, 2016). Our considerations are based on the properties of polars of plane singularities in characteristic p.

Dedicated to Antonio Campillo on the occasion of his 65th birthday

Appendix

Let $\overrightarrow w=(n,m)\in ({\mathbf {N}}_+)^2$ be a weight.

Lemma A.1

Let f, g ∈K[[x, y]] be power series without constant term. Then

$$\displaystyle \begin{aligned} i_0(f,g)\geq \frac{\left(\mathrm{ord}_{\overrightarrow w}(f) \right) \left(ord_{\overrightarrow w}(g) \right)}{mn},\end{aligned}$$

with equality if and only if the system of equations

$$\displaystyle \begin{aligned} \left\{\begin{array}{l} \mathrm{in}_{\overrightarrow w}(f)=0,\\ \\ \mathrm{in}_{\overrightarrow w}(g)=0 \end{array} \right. \end{aligned}$$

has the only solution (x, y) = (0, 0).

Proof

By a basic property of the intersection multiplicity (see for example [19, Proposition 3.8 (v)]) we have that for any nonzero power series $\tilde f, \tilde g$

$$\displaystyle \begin{aligned} i_0(\tilde f, \tilde g)\geq \mathrm{ord}(\tilde f) \mathrm{ord}(\tilde g), \end{aligned} $$

(7)

with equality if and only if the system of equations $\mathrm {in}(\tilde f)=0$, $\mathrm {in}(\tilde g)=0$ has the only solution (0, 0). Consider the power series $\tilde f(u,v)=f(u^n,v^m)$ and $\tilde g(u,v)=g(u^n,v^m)$. Then $i_0(\tilde f, \tilde g)=i_0(f,g)i_0(u^n,v^m)=i_0(f,g)nm$, $\mathrm {ord}(\tilde f)=\mathrm {ord}_{\overrightarrow w}(f)$, $\mathrm {ord}(\tilde g)=\mathrm {ord}_{\overrightarrow w}(g)$ and the lemma follows from (7).

Lemma A.2

Let f ∈K[[x, y]] be a non-zero power series. Then

$$\displaystyle \begin{aligned} i_0\left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)\geq \left(\frac{\mathrm{ord}_{\overrightarrow w}(f)}{n}-1\right)\left(\frac{\mathrm{ord}_{\overrightarrow w}(f)}{m}-1\right) \end{aligned}$$

with equality if and only if f is a semi-quasihomogeneous singularity with respect to ${\overrightarrow w}$.

Proof

The following two properties are useful:

$$\displaystyle \begin{aligned} \mathrm{ord}_{\overrightarrow w} \left(\frac{\partial f}{\partial x}\right)\geq \mathrm{ord}_{\overrightarrow w}(f)-n \;\;{\text{with equality if and only if }} \; \frac{\partial }{\partial x} \mathrm{in}_{\overrightarrow w}(f)\neq 0, \end{aligned} $$

(8)

$$\displaystyle \begin{aligned} {\text{if}} \; \frac{\partial }{\partial x} \mathrm{in}_{\overrightarrow w}(f)\neq 0 \;{\mathrm{then}} \; \mathrm{in}_{\overrightarrow w} \left(\frac{\partial f}{\partial x}\right)=\frac{\partial }{\partial x} \mathrm{in}_{\overrightarrow w}(f). \end{aligned} $$

(9)

By the first part of Lemma A.1 and Property (8) we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} i_0\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)&\displaystyle \geq &\displaystyle \frac{\left(\mathrm{ord}_{\overrightarrow w}\left( \frac{\partial f}{\partial x}\right) \right) \left(ord_{\overrightarrow w} \left(\frac{\partial f}{\partial y} \right)\right)}{nm}\geq \frac{\left(\mathrm{ord}_{\overrightarrow w}(f)-n \right) \left(ord_{\overrightarrow w}(f)-m \right)}{nm}\\ &\displaystyle =&\displaystyle \left(\frac{\mathrm{ord}_{\overrightarrow w}(f)}{n}-1\right)\left(\frac{\mathrm{ord}_{\overrightarrow w}(f)}{m}-1\right). \end{array} \end{aligned} $$

Using the second part of Lemma A.1 and Properties (8) and (9) we check that $i_0\left (\frac {\partial f}{\partial x},\frac {\partial f}{\partial y}\right )= \left (\frac {\mathrm {ord}_{\overrightarrow w}(f)}{n}-1\right )\left (\frac {\mathrm {ord}_{\overrightarrow w}(f)}{m}-1\right )$ if and only if f is a semi-quasihomogeneous singularity with respect to ${\overrightarrow w}$.

Lemma A.3 (Hensel’s Lemma [13, Theorem 16.6])

Suppose that $\mathrm {in}_{\overrightarrow w}(f)=\psi _1\cdots \psi _s$ with pairwise coprime ψ _i . Then f = g ₁⋯g _s ∈K[[x, y]] with $\mathrm {in}_{\overrightarrow w} (g_i)=\psi _i$ for i = 1, …, s.