Milnor–Tjurina conjecture for non-complete-intersection surface singularities in C4

Published counterexample to the equality characterization

The conjecture as stated is false: equality $\mu=\tau-1$ does not imply quasi-homogeneity. The following family of counterexamples was constructed by Anne Frühbis-Krüger.

For integers $k,\ell\geq 3$, let $(X_{k,\ell},0)\subset(\mathbb C^4,0)$ be defined by the maximal minors of

$$A_{k,\ell}= \begin{pmatrix} z+x & y & x^k+w^2\\ w^\ell & z & y \end{pmatrix}.$$

This is an isolated Cohen–Macaulay codimension-two surface singularity of type $(2,3,2)$. In particular it is a normal surface germ. Its determinantal ideal has height two and three minimal generators, so it is not a complete intersection.

Let $Y_{k,\ell}$ be its Tjurina transform. It has precisely two isolated singularities:

$$(Y_{k,\ell},p_0)\cong A_{\ell-1}, \qquad (Y_{k,\ell},p_\infty)\cong D_{k+1}.$$

Both are quasi-homogeneous. Consequently,

$$\sum_{p\in\operatorname{Sing}(Y_{k,\ell})}\mu(Y_{k,\ell},p) =(\ell-1)+(k+1)=k+\ell.$$

For an isolated determinantal surface of type $(2,3,2)$, the topology of the Tjurina transform gives

$$\mu(X_{k,\ell}) = 1+\sum_{p\in\operatorname{Sing}(Y_{k,\ell})} \mu(Y_{k,\ell},p),$$

and hence

$$\mu(X_{k,\ell})=k+\ell+1.$$

On the deformation-theoretic side,

$$\tau(X_{k,\ell}) = h^1(Y_{k,\ell},T_{Y_{k,\ell}}) + \sum_{p\in\operatorname{Sing}(Y_{k,\ell})} \tau(Y_{k,\ell},p).$$

For this family, the direct $T^1$-calculation gives

$$h^1(Y_{k,\ell},T_{Y_{k,\ell}})=2$$

and therefore

$$\tau(X_{k,\ell}) = 2+(\ell-1)+(k+1) = k+\ell+2.$$

Thus

$$\boxed{\mu(X_{k,\ell})=\tau(X_{k,\ell})-1}.$$

Nevertheless, $X_{k,\ell}$ is not quasi-homogeneous. One way to see the incompatibility is to compare the two hyperplane sections used in the cited construction. The section $w=0$ requires weights proportional to

$$(\operatorname{wt}x,\operatorname{wt}y,\operatorname{wt}z) =(2,k+1,2),$$

whereas the section $x=0$ requires

$$(\operatorname{wt}y,\operatorname{wt}z,\operatorname{wt}w) =(\ell+4,2\ell+2,3).$$

In particular, a common grading would require simultaneously

$$\frac{\operatorname{wt}y}{\operatorname{wt}z} = \frac{k+1}{2} \qquad\text{and}\qquad \frac{\operatorname{wt}y}{\operatorname{wt}z} = \frac{\ell+4}{2\ell+2},$$

which is impossible for $k,\ell\geq3$.

For the smallest example $k=\ell=3$,

$$A_{3,3}= \begin{pmatrix} z+x & y & x^3+w^2\\ w^3 & z & y \end{pmatrix}, \qquad \mu(X_{3,3})=7,\quad \tau(X_{3,3})=8,$$

while $X_{3,3}$ is not quasi-homogeneous.

Therefore the “equality if and only if quasi-homogeneous” clause of the conjecture is disproved. This family does not refute the remaining inequality $\mu\geq\tau-1$.

Reference. A. Frühbis-Krüger, “On discriminants, Tjurina modifications and the geometry of determinantal singularities,” Topology and its Applications 234 (2018), 375–396, Section 4, especially pp. 21–23; arXiv:1611.02625.