Corrigendum to "Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities"

TL;DR

This paper corrects a previous mistake regarding the conductor at multiplanar isolated singularities, clarifying the accurate statement without affecting the validity of earlier results.

Contribution

It provides an explicit correction to a prior statement about the conductor at multiplanar singularities, ensuring clarity and accuracy in the mathematical description.

Findings

Corrected the statement about the conductor at multiplanar singularities

Ensured previous results remain valid despite the correction

Clarified the mathematical understanding of the singularity structure

Abstract

We explicitly fix a mistake in a preliminary statement of our previous paper on the conductor at a multiplanar singularity. The correction is not immediate and, though the mistake does not affect correctness of the subsequent results, the wrong statement could easily be misleading.

Equations

$$G\left(\overline{A}\right)$$

$${\nu }_n$$

$$\mathrm{Proj}\left(G\left(A\right)\right)\cong \mathrm{Proj}\left(G_{\overline{\mathfrak{m}}}\left(\overline{A}\right)\right).$$

$$f^{n_0}\in {\overline{\mathfrak{m}}}^{n{n}_0+n_0}={\mathfrak{m}}^{n{n}_0+n_0}\overline{A}={\mathfrak{m}}^{n{n}_0}{\overline{\mathfrak{m}}}^{n_0}\subseteq {\mathfrak{m}}^{n{n}_0+1},$$

$$\sum _{i=n_0}^{n-1}{\dim }_k{G}_{\overline{\mathfrak{m}}}{\left(\overline{A}\right)}_i=l\left(\frac{{\overline{\mathfrak{m}}}^{n_0}}{{\overline{\mathfrak{m}}}^n}\right)\le l\left(\frac{\mathfrak{m}}{{\mathfrak{m}}^n}\right)=\sum _{i=1}^{n-1}{\dim }_k{G\left(A\right)}_i.$$

$$\sqrt{\mathfrak{b}}=\mathfrak{m}$$

$$k[X_0,\dots ,X_r]\to G\left(A\right)\stackrel{\nu }{⟶}G\left(\overline{A}\right)\stackrel{\sim }{\to }\prod _{i=1}^e{G}_{{\mathfrak{a}}_i}\left(k[t,s]\right)\stackrel{{\pi }_i}{⟶}k\left[\overline{t-a_{i1}},{\overline{s}}_i\right].$$

$$Y:=\mathrm{Proj}\left(G\left(A\right)\right)$$

Full text

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Proof. Since $A$ is noetherian and $\sqrt{\mathfrak{b}}=\mathfrak{m}$, we have $\mathfrak{m}^{n_{0}}\subseteq\mathfrak{b}$ for some $n_{0}$, hence $\overline{\mathfrak{m}}^{{n_{0}}}=\mathfrak{m}^{n_{0}}\overline{A}\subseteq\mathfrak{b}\overline{A}=\mathfrak{b}\subseteq\mathfrak{m}$.

If the class $x$ of a $f\in\mathfrak{m}^{n}\smallsetminus\mathfrak{m}^{n+1}$ in $\operatorname{G}\left(A\right)_{n}$ is in the kernel of $\nu_{n}$ that is, $f$ belongs to $\overline{\mathfrak{m}}^{n+1}$, we have

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hence $x^{n_{0}}=0\in\operatorname{G}\left(A\right)_{nn_{0}}$, that is, $x$ is nilpotent. Since $\operatorname{Proj}\left(\operatorname{G}\left(A\right)\right)$ is reduced, there exists $n_{1}$ such that for all $n\geq{n_{1}}$ there is no nilpotent $x\in\operatorname{G}\left(A\right)_{n}\smallsetminus\{0\}$, and therefore $\nu_{n}$ is injective for all $n\geq n_{1}$.

Let us consider the powers $\mathfrak{m}^{n}$ and $\overline{\mathfrak{m}}^{n}$ as $A$-modules, and denote by $l(M)$ the length of an arbitrary $A$-module $M$. Note also that the graded components $\operatorname{G}_{\overline{\mathfrak{m}}}\left(\overline{A}\right)_{n}=\overline{\mathfrak{m}}^{n}/\overline{\mathfrak{m}}^{n+1}$ are vector spaces over the residue field $k:=A/\mathfrak{m}$, because $\overline{\mathfrak{m}}^{n+1}=\mathfrak{m}\overline{\mathfrak{m}}^{n}$; the same is obviously true for the graded components of $\operatorname{G}\left(A\right)$. For every $n\geq n_{0}$ we have $\mathfrak{m}^{n}\subseteq\overline{\mathfrak{m}}^{n}\subseteq\overline{\mathfrak{m}}^{n_{0}}\subseteq\mathfrak{m}$, hence

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Note that $\dim_{k}\operatorname{G}_{\overline{\mathfrak{m}}}\left(\overline{A}\right)_{i}\geq\dim_{k}{\operatorname{G}\left(A\right)}_{i}$ for all $i\geq n_{1}$, because $\nu_{i}$ is injective for that values. Suppose now that it is not true that $\nu_{n}$ is an isomorphism for all $n>>0$. Then we can find as many values of $i$ as we want for which $\dim_{k}\operatorname{G}_{\overline{\mathfrak{m}}}\left(\overline{A}\right)_{i}>\dim_{k}{\operatorname{G}\left(A\right)}_{i}$. This contradicts (3) for a sufficiently large $n$.

Finally, it is well known that if a graded ring homomorphism $S\to T$ preserving degrees induces isomorphisms on all components of sufficiently large degrees, then it induces an isomorphism $\operatorname{Proj}\left(T\right)\overset{\sim}{\to}\operatorname{Proj}\left(S\right)$. $\,{}_{\Box}$

Next, [1, Remark 2.2] is invoked in [1, Section 5, p. 2977, lines 20–21]:

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At that point, we do not have that $\operatorname{Proj}\left(\operatorname{G}\left(A\right)\right)$ is reduced, and this fact is part of n. 3 of Claim 5.1 (the only point of that Claim that was still to be proved). In what follows we assume the notation of [1, Section 5]. To fix the mistake, the sentence (4) above must be dismissed, but we can keep the fact that $\mathfrak{J}=\mathfrak{m}\overline{A}$ (which is true, because it had been proved earlier that $\mathfrak{J}=\mathfrak{m}B$ and $\overline{A}=B$).

Let us look at the subsequent discussion in [1]. First of all, the isomorphism $\operatorname{G}\left(\overline{A}\right)\cong\prod_{i=1}^{e}\operatorname{G}_{\mathfrak{a}_{i}}\left(k[t,s]\right)$ is pointed out. Here, let us also denote by $\pi_{i}:\operatorname{G}\left(\overline{A}\right)\to\operatorname{G}_{\mathfrak{a}_{i}}\left(k[t,s]\right)=k\left[\overline{t-a_{i1}},\overline{s}_{i}\right]$ the projection on the $i$th factor. Let us split as follows the natural homomorphism displayed at [1, p. 2977, line 25]:

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(in the description $X_{j}\mapsto\overline{x_{j}}$ given in the paper, $\overline{x_{j}}$ denotes the class of $x_{j}\in R\subset k[t,s]$ in the degree one component of $\operatorname{G}_{\mathfrak{a}_{i}}\left(k[t,s]\right)$). Assuming that $X_{0},\ldots,X_{r}$ act as the coordinate functions on $k^{r+1}$, and the linear forms in $k[X_{0},\ldots,X_{r}]$ act accordingly, for each choice of distinct $i_{0},i_{1}\in\left\{1,\ldots,e\right\}$, since $l_{i_{0}}$ and $l_{i_{1}}$ are skew lines we can fix linear forms $T_{i_{0}i_{1}}$, $S_{i_{0}i_{1}}$ such that

• •

$T_{i_{0}i_{1}}$ vanishes on $\mathbf{a}_{i_{0}}$, $\mathbf{b}_{i_{0}}$, $\mathbf{b}_{i_{1}}$, and takes value $1/\rho_{i_{1}}$ on $\mathbf{a}_{i_{1}}$, with $\rho_{i_{1}}$ being as in [1, Footnote 4];

• •

$S_{i_{0}i_{1}}$ vanishes on $\mathbf{a}_{i_{0}}$, $\mathbf{b}_{i_{0}}$, $\mathbf{a}_{i_{1}}$, and takes value $1$ on $\mathbf{b}_{i_{1}}$.

Taking into account [1, Footnote 4], the images of $T_{i_{0}i_{1}}$ and $S_{i_{0}i_{1}}$ in $k\left[\overline{t-a_{i1}},\overline{s}_{i}\right]$ through (5) vanish for $i=i_{0}$, and equal $\overline{t-a_{i1}}$ and $\overline{s}_{i}$, respectively, for $i=i_{1}$. It follows that the image of $T_{1i}\cdots T_{(i-1)i}T_{(i+1)i}\cdots T_{ei}$ in $\prod_{i=1}^{e}\operatorname{G}_{\mathfrak{a}_{i}}\left(k[t,s]\right)$ through (5) is $\left(0,\ldots,\overline{t-a_{i1}}^{e-1},0,\ldots,0\right)$, where the nonzero component occurs at the $i$-th place. In a similar way, every element of the form $\left(0,\ldots,\overline{t-a_{i1}}^{a}\overline{s}^{b},0,\ldots,0\right)$, with $a+b\geq e-1$, is the image of a product of $a+b$ linear forms, suitably chosen among the $T_{i_{0}i_{1}}$s and the $S_{i_{0}i_{1}}$s. It follows that the homomorphism $k[X_{0},\ldots,X_{r}]\to\prod_{i=1}^{e}\operatorname{G}_{\mathfrak{a}_{i}}\left(k[t,s]\right)$ is surjective in degrees $\geq e-1$. Hence $\nu_{d}$ is surjective for $d\geq e-1$, and is in fact an isomorphism by [1, Lemma 3.1]. Thus $\nu$ induces the required isomorphism $\operatorname{Proj}\left(\operatorname{G}\left(A\right)\right)\cong\operatorname{Proj}\left(\operatorname{G}\left(\overline{A}\right)\right)$.

Finally, let us take this occasion to also fix a few typos and mild issues.

In the summarized description of the main result [1, Theorem 3.2] given in [1, Introduction] at the beginning of p. 2970, it is missed the hypothesis (duly reported in the statement of the theorem and in the abstract) that $\operatorname{Proj}\left(\operatorname{G}\left(A\right)\right)$ must be reduced. 2. 2.

Typo at [p. 2971, lines 12–14]: “$\mathbb{P}^{r+1}:=\operatorname{Proj}\left(k[X_{1},\ldots,X_{k}]\right)\ldots$ subscheme of $\mathbb{P}^{r+1}$” should be replaced with “$\mathbb{P}^{r}:=\operatorname{Proj}\left(k[X_{0},\ldots,X_{r}]\right)\ldots$ subscheme of $\mathbb{P}^{r}$”. 3. 3.

At the beginning of the proof of [1, Theorem 4.1] one finds

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Since $\operatorname{Proj}\left(\operatorname{G}\left(A\right)\right)$ is reduced, Proposition 1 can serve as a replacement for [1, Remark 2.2] in this situation, too. 4. 4.

In [1, Section 5] (main example), $e$ must be assumed $\geq 2$ and $n$ has to be replaced with $r$ in some lists such as $x_{0},\ldots,x_{n}$, $f_{2},\ldots,f_{n}$, $h_{2},\ldots,h_{n}$ and $X_{0},\ldots,X_{n}$, as well as in the exponent $n+1$ in [1, Equation (6)].

In the sentence below Equation (6), ‘such that $g(\overline{t})$ takes one of the above mentioned special values’ should be completed with ‘or $g^{\prime}(\overline{t})=0$’. Later, ‘vanishes in the ring $T:=B_{f}\otimes_{A_{f}}B_{f}$’ should be ‘vanishes over the ring $T:=B_{f}\otimes_{A_{f}}B_{f}$’ (that is, each component of the $(n+1)$-tuple under consideration vanishes in $T$).