Tangent cone of a non-isolated singular point

Question

Assume $X\subset \mathbb P^n$ is a variety (Edit: let's say $X$ is a hypersurface in $\mathbb P^n$, as pointed out in the comment) and $x\in X$ is a singular point which is not isolated. Intuitively, I think the quadratic tangent cone at $x$ (i.e. the $Z(f_2)$ where $f=f_2+f_3+ \cdots$ is the defining equation of $X$ on an affine chart centered at $x$) is always degenerate, as it should degenerate "along" the singular locus. Is this true? And is there any reference about this?

Thanks!

Answer 1

When $X$ is a hypersurface, it reduces to consider a holomorphic function $f:\mathbb C^n\to \mathbb C$. Let $x$ be a singular point on $X:=f^{-1}(0)$. Then this theorem is what you really need:

Theorem: $x$ is an isolated singularity on $X$ if and only if its Milnor number $\mu_x< \infty$.

Here the Milnor number at $x$ is defined to be $\mu_x:=\dim_{\mathbb C} \mathcal{O}_{X,x}/J$, with $J$ the Jacobi ideal

$$J:=\langle\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n}\rangle.$$

Assuming the theorem is true, now it is an exercise to show:

Claim: The quadric tangent cone at $x$ is nondegenerate if and only if $x$ is an ordinary node ($\mu_x=1$). In particular, a non-isolated singularity must has degenerate quadric tangent cone.

Proof of Theorem: If $x$ is isolated, then (up to choosing a smaller affine chart) the common zero locus of $J$ is the single point $\{x\}$, therefore by Nullstellensatz $\sqrt{J}=\mathfrak{m}_x$. Since $\mathfrak{m}_x$ is finitely generated, there exists an integer $r\ge 1$, s.t., $\mathfrak{m}_x^r\subset J$. Now $$\dim_{\mathbb C}\mathcal{O}_{X,x}/\mathfrak{m}^r_x=\sum_{i=1}^r\dim_{\mathbb C}\mathfrak{m}_x^{i-1}/\mathfrak{m}_x^{i}<\infty$$ implies that $\mu_x$ is finite. The other direction needs Nakayama's lemma. $\square$

You can find the literature of the Theorem in Chapter 1, Lemma 2.3 (together with Corollary 1.74) in Introduction to Singularities and Deformations by Greuel, Lossen and Shustin.

I should also point it out that there is a topological proof by Milnor in Appendix B of his famous book Singular Points of Complex Hypersurfaces, where he studied topology of isolated hypersurface singularity in 1968 and introduced $\mu_x$ (in a topological way).

Topological implication of Milnor number $\mu_x$: This is irrelavent to OP's question, but I'd like to remark here that if $x$ is an isolated singularity, and $B$ a small ball around $x$ in $\mathbb C^n$, then for $t\neq 0$ and $|t|<<1$, the Milnor fiber $f^{-1}(t)\cap B$ has topological type bouquet of spheres $$\underbrace{S^{n-1}\vee\cdots \vee S^{n-1}}_{\mu_x}$$

In other words, $\mu_x$ is the number of $(n-1)$-spheres that appear through a "perturbation" of the isolated singularity.