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This problem is from Greuel et al., Introduction to Singularities and Deformations.


Determine the singular locus of the complex spaces defined by the following $\mathcal{O}_{\mathbb{C}^n}$-ideals:

(a) $\langle (x_1^2+x_2^2)^2-x_1^2+x_2^2\rangle\subset \mathcal{O}_{\mathbb{C}^2}$(Bernoulli's lemniscate)

(b) $\langle x_1^2-x_2^2x_3\rangle\subset \mathcal{O}_{\mathbb{C}^2}$ (Whitney's umbrella)

(c) $\langle x_1x_2,x_2x_3,x_1x_3\rangle\subset \mathcal{O}_{\mathbb{C}^3}$ (coordinate cross)

(d) $\langle x_1x_3,x_2x_3\rangle\subset \mathcal{O}_{\mathbb{C}^3}$


I'm willing to use "Rank theorem", whose statement is as follows:

Let $X$ be a complex space, $x\in X$, and let $\mathcal{O}_{X,x}\cong \mathbb{C}\{x_1,\ldots,x_m\}/I$ where $I=\langle f_1,\ldots,f_k\rangle$. Then the following conditions are equivalent:

(a) $(X,x) $ is regular and $\dim(X,x)=n$.

(b) $\mathcal{O}_{X,x}\cong \mathbb{C}\{x_1,\ldots,x_n\}$

(c) There is an open subset $U\subset X$, $x\in U$, such that $(U,\mathcal{O}_X|_U)$ is a complex manifold of dimension $n$.

(d) There is an open neighborhood $D$ of $\mathbf{0}$ in $\mathbb{C}^m$ such that the $f_i$ converge in $D$ and $$\text{rank}\left(\frac{\partial f_i}{\partial x_j}(\mathbf{p})\right)_{(i,j)}=m-n$$ for all $\mathbf{p}\in D$.

Here are my questions:

  1. How should I describe $\mathcal{O}_{X,x}$ at each $x\in X$?

  2. Intuitively, I think that the point where the derivatives of all $f_i$'s vanish is exactly the singular locus. Am I right? If not, how should I find the singular locus?

I appreciate your help in advance!

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  1. It's not really so easy to say beyond it's the ring of germs of holomorphic functions at that point. Even describing explicitly what $\Bbb C \{x_1,x_2,\cdots,x_n\}$ is can be a little dicey if you want to get really specific - can you explicitly pick out all the convergent power series in multiple variables?

  2. This is a great idea - you're almost there, except that it's not just where the derivatives all vanish, but where the matrix of partial derivatives of the defining equations is less than full rank. This is typically called the Jacobian criterion, and is part (d) of the "rank theorem" you've quoted. To see an example where you really need this, look at example (d). Here the matrix of partial derivatives of the defining equations is

$$\begin{pmatrix} x_3 & 0 & x_1 \\ 0 & x_3 & x_2 \end{pmatrix}$$

which has full rank iff $x_3\neq 0$, which says that the entire $x_3$ plane is singular, despite the point $(1,1,0)$ not having all partial derivatives of the defining equations vanish there. In general, you'll want to write down the matrix of partial derivatives of the defining equations and look for where it's singular.

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