# Relationship between Hilbert-Samuel multiplicity and polar multiplicity

Let $$f \in \mathbb{C}[[x,y]]$$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $$e_f$$ of $$f$$ is given as follows: $$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \dim_{\mathbb{C}} \mathbb{C}[[x,y]]\bigg/\left(f, \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)^s\right)$$ I've noticed that in various examples (like the $$A_{n-1}$$-singularities $$f = y^2 - x^n$$, or more generally, the hypercuspidal singularities $$f = y^a - x^b$$) that $$e_f$$ is equal to the intersection multiplicity of $$f$$ and a generic polar'' of $$f$$. More precisely, I've observed that the following equality holds in many examples: $$e_f = \dim_{\mathbb{C}} \mathbb{C}[[x,y]]\bigg/\left(f, \alpha \cdot \frac{ \partial f}{\partial x} - \beta \cdot \frac{\partial f}{ \partial y}\right) \qquad (*)$$ where $$[\alpha : \beta] \in \mathbb{P}_{\mathbb{C}}^1$$ is generic. My question: is it well-known (or obvious) whether the equality $$(*)$$ holds for an arbitrary isolated plane curve singularity germ $$f$$?

(Note: the colength of the polar (the quantity on the right of $$(*)$$) is of interest because it is related to Teissier's notion of polar invariant.'')

What I have so far: Note that the polar is itself a complete intersection, so the colength of the polar in the ring $$\mathbb{C}[[x,y]]/(f)$$ is equal to the Hilbert-Samuel multiplicity of the polar. Since the Hilbert-Samuel multiplicity of an ideal is equal to that of any reduction, it suffices to show that the ideal $$I = \left(\alpha \cdot \frac{\partial f}{\partial x} - \beta \cdot \frac{\partial f}{\partial y}\right)$$ generated by the polar is a reduction of the ideal $$J = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$$ (here, we are regarding $$I,J$$ as ideals of the ring $$\mathbb{C}[[x,y]]/(f)$$). I.e., it suffices to show that for some $$s \geq 0$$, the obvious inclusion $$I(J^s) \subset J^{s+1}$$ of ideals of $$\mathbb{C}[[x,y]]/(f)$$ is in fact an equality. Now restrict to the case where $$f \in \mathbb{C}[x,y]$$ is homogeneous of degree $$d$$. Then it is not hard to check by counting generators that $$IJ^{d-1} = (x,y)^{d(d-1)} \supset J^d$$ (as ideals of $$\mathbb{C}[[x,y]]/(f)$$). I do not currently know how to generalize this argument to handle germs $$f$$ that are not homogeneous.

First, the quantity you defined is the Hilbert-Samuel multiplicity of the ideal $$J= (f_x,f_y)$$ in $$R=\mathbb C[[x,y]]/(f)$$. The multiplicity of $$f$$ usually refers to the multiplicity of the maximal ideal $$m$$ of $$R$$.

As you noted, it is enough to show that a generic combination of the generators of $$J$$ is a reduction. This is true much more generally, and basically the point is to consider the fiber cone $$S= F(I) = k\oplus \frac {J}{mJ}\oplus \frac {J^2}{mJ^2}\oplus...$$. This algebra has dimension $$1$$, and is generated in degree one. Then, for a generic linear form $$l \in S_1$$, $$S/lS$$ is $$0$$-dimensional, so $$lS_n=S_{n+1}$$ for $$n\gg 0$$. But since $$l \in S_1= J/mJ$$, this says exactly that $$J^nc = J^{n+1}$$ where $$c$$ is a lift of $$l$$.