Local tropicalizations of splice type surface singularities

With an appendix by Jonathan Wahl (Department of Mathematics, The University of North Carolina at Chapel Hill, 120 E Cameron Avenue, CB #3250, Chapel Hill, NC 27599-3250, USA. Email: jmwahl@email.unc.edu)

Abstract

Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham–Brieskorn–Hamm complete intersections of dimension two. Their construction depends on a weighted tree called a splice diagram. In this paper, we study these singularities from the tropical viewpoint. We characterize their local tropicalizations as the cones over the appropriately embedded associated splice diagrams. As a corollary, we reprove some of Neumann and Wahl’s earlier results on these singularities by purely tropical methods, and show that splice type surface singularities are Newton non-degenerate complete intersections in the sense of Khovanskii. We also confirm that under suitable coprimality conditions on its weights, the diagram can be uniquely recovered from the local tropicalization. As a corollary of the Newton non-degeneracy property, we obtain an alternative proof of a recent theorem of de Felipe, González Pérez and Mourtada, stating that embedded resolutions of any plane curve singularity can be achieved by a single toric morphism, after re-embedding the ambient smooth surface germ in a higher-dimensional smooth space. The paper ends with an appendix by Jonathan Wahl, proving a criterion of regularity of a sequence in a ring of convergent power series, given the regularity of an associated sequence of initial forms.

Appendix A. Initial ideals and local regular sequences

In [38], the authors invoke a folklore lemma in commutative algebra in order to prove several of their main theorems. This result involves regular sequences in a polynomial ring and their initial forms with respect to integer weight vectors. As originally stated, [38, Lemma 3.3] is not quite-correct: the global setting must be replaced by a local one. This appendix provides a complete proof of this result in the local setting of convergent power series near the origin, a result we could not locate in the literature. This local version agrees with the general framework of [38]. Throughout, we let n be a positive integer and let denote the local ring of convergent power series ${\mathbb {C}}\{z_1,\ldots , z_n\}$ near the origin.

We start by stating our main result, namely, a reformulation of [38, Lemma 3.3] in the local setting. Its proof will be given at the end of this appendix, after discussing a series of preliminary technical results. Note that the same statement and proof will hold if ${\mathcal {O}}$ denotes the localization of the polynomial ring ${\mathbb {C}}[z_1, \ldots , z_n]$ at the maximal ideal of the origin of ${\mathbb {C}}^n$.

Theorem A.1

Let $(f_1,\ldots , f_s)$ be a finite sequence of elements in the maximal ideal ${\mathfrak {m}}$ of ${\mathcal {O}}$, and let J be the ideal generated by them. Fix a positive weight vector $w_{}\in ({\mathbb {Z}}_{>0})^n$. Assume that $({\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_s))$ is a regular sequence in ${\mathcal {O}}$. Then:

• 1. the sequence $(f_1,\ldots , f_s)$ is also regular, and

• 2. the $w_{}$-initial ideal ${\text {in}}_{w_{}}(J){\mathcal {O}}$ is generated by $\{{\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_s)\}$.

Remark A.2

As mentioned earlier, Theorem A.1 does not hold in the polynomial setting. For instance, $(z_1(1-z_1), z_2(1-z_1))$ is a regular sequence in the local ring ${\mathbb {C}}\{z_1,z_2\}$ but not in the polynomial ring ${\mathbb {C}}[z_1,z_2]$. However, the sequence $(z_1,z_2)$ of initial forms with respect to any weight vector $w_{}\in ({\mathbb {Z}}_{>0})^2$ is regular in both rings.

Remark A.3

The regularity of the sequence of $w_{}$-initial forms is needed in Theorem A.1. As an example, fix $n=4$, $w_{}=(1,1,1,1)$, and consider the sequence $(f_1,f_2)$ with

$$\begin{aligned} f_1:=z_1^2+z_2^4-z_3^3 \qquad \text { and } \qquad f_2:=z_1\,z_2-z_4^3. \end{aligned}$$

By construction, $(f_1,f_2)$ is a regular sequence in ${\mathcal {O}}$ defining an isolated complete intersection surface singularity. The sequence of initial forms $({\text {in}}_{w_{}}(f_1), {\text {in}}_{w_{}}(f_2)) = (z_1^2, z_1\,z_2)$ is not regular, and the $w_{}$-initial ideal of $(f_1,f_2){\mathcal {O}}$ is generated by ${\text {in}}_{w_{}}(f_1)$, ${\text {in}}_{w_{}}(f_2)$ and ${\text {in}}_{w_{}}(z_2\,f_1 -z_1\,f_2) = - z_2\,z_3^3 + z_1\,z_4^3$.

Throughout, we fix $w_{}\in ({\mathbb {Z}}_{>0})^n$ and an arbitrary sequence $(f_1,\ldots , f_s)$ of elements of the maximal ideal ${\mathfrak {m}}$. We let J be the ideal generated by the $f_i$’s. Consider the first few steps in the Koszul complex of ${\mathcal {O}}$-modules determined by it (see, e.g., [51, Sect. IV.A]):

(A.1)

The map $d_1:E\rightarrow {\mathcal {O}}$ sends $e_i$ to $f_i$ for each $i=1,\ldots , s$ and the kernel of $d_1$ is the module of relations between the given generators of J. The morphism $d_2:F\rightarrow E$ sends $e_{ij}$ to $f_j\,e_i-f_i\,e_j$, and its image is the submodule of “trivial relations” between $(f_1,\ldots , f_s)$. By definition, the image of $d_2$ lies in R, so we view $d_2$ also as as a map $d_2:F\rightarrow R$.

By a standard result in commutative algebra (see, e.g., [51, Proposition 3, Chapter IV.A.2]) we have:

Proposition A.4

The sequence $(f_1,\ldots ,f_s)$ of elements in ${\mathfrak {m}}$ is regular in ${\mathcal {O}}$ if and only if the Koszul complex (A.1) is exact at E.

Since the definition of E does not depend on the order of the sequence $(f_1,\ldots , f_s)$, the following consequence arises naturally:

Corollary A.5

If $(f_1,\ldots , f_s)$ is a regular sequence in ${\mathcal {O}}$, any reordering of it is also a regular sequence.

The weight vector $w_{}$ inducing the $w_{}$-weight valuation (3.1) on ${\mathcal {O}}$ endows this ring with a weight filtration by ideals $(I_p)_{p\ge 0}$, where . Similarly, we can filter ${\mathcal {O}}$ via the ideals $({\mathfrak {m}}^p)_{p\ge 0}$. Both filtrations are cofinal since

$$\begin{aligned} I_{d p} \subseteq {\mathfrak {m}}^p \subseteq I_p \quad \text { for all }\; p\ge 0, \end{aligned}$$

(A.2)

where is the maximum among all coordinates of $w_{}$. It follows from this that the completions of ${\mathcal {O}}$ with respect to both filtrations are canonically isomorphic. The completion induced by the ${\mathfrak {m}}$-adic filtration $({\mathfrak {m}}^p)_{p\ge 0}$ is the ring of formal power series in n variables.

In a similar fashion, we can filter the modules E and F appearing in (A.1) via $(E_p)_{p\ge 0}$ and $(F_p)_{p\ge 0}$, respectively, by assigning the weights $w_{}(f_i)$ and $w_{}(f_i)+w_{}(f_j)$ to $e_i$ and $e_{ij}$, respectively. More precisely,

(A.3)

These choices ensure that the maps $d_1$ and $d_2$ from the Koszul complex (A.1) preserve the filtration. In addition, the module R of relations is filtered as well, via

(A.4)

We use these filtrations to define the $w_{}$-initial forms on E and F. We state the definition for E, since the one for F is analogous. The definition for R is given by restriction.

Definition A.6

Given any $g\in E$ with $g \ne 0$, we let p be the unique integer such that $g\in E_{p}\smallsetminus E_{p+1}$. An element $g:=\sum _{i=1}^s r_i \, e_i \in E_p\smallsetminus E_{p+1}$ satisfies $w_{}(r_i) + w_{}(f_i) \ge p$ for all $i\in \{1,\ldots , s\}$ and equality must hold for some index i. Let I be the set of indices where equality is achieved. The $w_{}$-initial form of g is . We set ${\text {in}}_{w_{}}(0) = 0$.

By Proposition A.4, the regularity of the sequence $(f_1,\ldots , f_s)$ is equivalent to the surjectivity of the map $d_2:F\rightarrow R$ induced by (A.1). We prove the latter in Lemma A.11, assuming the regularity of the sequence of $w_{}$-initial forms of all $f_i$’s.

Our first two lemmas use the regularity assumptions for the sequence of $w_{}$-initial forms to prove the surjectivity of $d_2:F\rightarrow R$ by working with the filtrations of F and R described above.

Lemma A.7

Assume that the sequence $({\text {in}}_w(f_1),\ldots , {\text {in}}_w(f_s))$ is regular in ${\mathcal {O}}$. Then, the morphism of ${\mathbb {C}}$-vector spaces $\varphi _p:F_p/F_{p+1}\rightarrow R_p/R_{p+1}$ induced by the morphism of ${\mathcal {O}}$-modules $d_2:F \rightarrow R$ is surjective for all integers $p \ge 0$.

Proof

We must show that modulo $R_{p+1}$, every element g of $R_p$ is the image of an element of $F_p/F_{p+1}$ under the map $\varphi _p$. If $g=0$, there is nothing to show, so we assume $g\ne 0$. In particular, g lifts to an element in $R_p\smallsetminus R_{p+1}$, which we denote by g as well. We write $g=\sum _{j=1}^s r_j\,e_j$.

Assume that ${\text {in}}_{w_{}}(g)$ has k many terms, with $k\in \{1,\ldots , s\}$ (see Definition A.6). By Corollary A.5, we can reorder the original sequence while preserving its regularity, and write ${\text {in}}_{w_{}}(g)$ as

$$\begin{aligned} {\text {in}}_{w_{}}(g)= \sum _{j=1}^k {\text {in}}_{w_{}}(r_{j}) e_{j}\quad \text { with }\quad w_{}(r_{j}) + w_{}(f_{j}) = p\quad \text { for all }\;j\in \{1,\ldots , k\}. \end{aligned}$$

We claim that g is congruent, modulo the image of $\varphi _p$, to an element of $R_p$ whose $w_{}$-initial form lies in the ideal generated by $\{{\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_{k-1})\}$. The original statement will follow by induction on $k\le s$.

Since $\sum _{j=1}^s r_j f_j = 0$ by definition of R and $w_{}(r_j)+w_{}(f_j)>p$ for $j\in \{k+1, \ldots , s\}$, we conclude that the expected $w_{}$-initial form of $\sum _{j=1}^s r_jf_j$ must vanish, i.e.,

$$\begin{aligned} \sum _{j=1}^k {\text {in}}_{w_{}}(r_{j}) {\text {in}}_{w_{}}(f_{j}) = 0. \end{aligned}$$

Therefore, ${\text {in}}_{w_{}}(r_{k}) {\text {in}}_{w_{}}(f_{k})$ is zero modulo the ideal $I:=\langle {\text {in}}_{w_{}}(f_1), \ldots , {\text {in}}_{w_{}}(f_{k-1})\rangle {\mathcal {O}}$. Since the sequence $({\text {in}}_{w_{}}(f_1), \ldots , {\text {in}}_{w_{}}(f_s))$ is regular, we conclude that ${\text {in}}_{w_{}}(r_{k})$ must lie in I.

Taking the $w_{}$-weight value of $r_{k}$ and each $f_{j}$ into account we write ${\text {in}}_{w_{}}(r_{k})$ as

$$\begin{aligned} {\text {in}}_{w_{}}(r_{k}) = \sum _{j=1}^{k-1} a_j {\text {in}}_{w_{}}(f_{j}), \end{aligned}$$

where $a_j$ is either 0 or a non-zero $w_{}$-weighted homogeneous polynomial with $w_{}(a_j)= p-w_{}(f_{k})-w_{}(f_{j})\ge 0$ for all $j\in \{1,\ldots , k-1\}$. It follows from this that the element

$$\begin{aligned} r_{k}':=r_{k}-\sum _{j=1}^{k-1} a_j f_j \end{aligned}$$

satisfies $w_{}(r_{k}')> p-w_{}(f_{k})$, so $r_{k}'\, e_{k} \in E_{p+1}$. Simple arithmetic manipulations give a new formula for g, i.e,

$$\begin{aligned} g = \sum _{j=1}^s r_j\, e_j = \sum _{j=1}^{k-1} (r_j + a_j\,f_{k}) \, e_j + \underbrace{\sum _{j=1}^{k-1} a_j(f_j\,e_{k} - f_{k} \,e_j)}_{=: \, h} + r_{k}'\, e_{k} + \sum _{j=k+1}^s r_j \,e_j. \end{aligned}$$

(A.5)

By construction, it follows that $h=\varphi _p(\sum _{j=1}^{k-1} a_j e_{jk}) \in \varphi _p(F_p/F_{p+1})$. Furthermore, ${\text {in}}_{w_{}}(g-h)$ only involves terms in the first of the four summands on the right-hand side of (A.5) since the last two summands lie in $E_{p+1}$. This establishes the claim. $\square$

Lemma A.8

Let J be the ideal of ${\mathcal {O}}$ generated by $\{f_1,\ldots , f_s\}$ and assume that $({\text {in}}_w(f_1),\ldots , {\text {in}}_w(f_s))$ is a regular sequence in ${\mathcal {O}}$. If $g\in J$ has $w_{}$-weight equal to $p \in {\mathbb {N}}$, then g admits an expression of the form $g=\sum _{i=1}^s a_if_i$, where $w_{}(a_i f_i)\ge p$ for all i. In particular, ${\text {in}}_{w_{}}(g)$ belongs to the ideal of ${\mathcal {O}}$ generated by $\{{\text {in}}_{w_{}}(f_1), \ldots , {\text {in}}_{w_{}}(f_s)\}$.

Proof

Since $g \in J$, we may write g as $g=\sum _{i=1}^s b_i\,f_i$ with $b_i\in {\mathcal {O}}$ for each i. Consider

$$\begin{aligned} p'=\min \{w_{}(b_i f_i): i=1,\ldots , s\}. \end{aligned}$$

Assume that this weight is achieved at k many terms, which we can fix to be $\{b_{1}\,f_{1}, \ldots , b_{k}\,f_{k}\}$ upon reordering. If $p'\ge p$ we have $w_{}(a_if_i)\ge p$ for all i and equality must hold for some i by definition of p. From here it follows that $p'=p$, so ${\text {in}}_{w_{}}(g) = \sum _{j=1}^k {\text {in}}_{w_{}}(a_{j}){\text {in}}_{w_{}}(f_{j})$, as we wanted to show.

On the contrary assume that $p'<p$. We claim that we can find an alternative expression $g=\sum _{j=1}^{s} b_j'\, f_j$ where the corresponding minimum weight $p'':=\min \{w_{}(b_j'\, f_j)\}$ satisfies $p''\ge p'$ and the number of summands realizing $p''$ is strictly smaller than k. An easy induction combined with the fact that $p',p\in {\mathbb {Z}}_{\ge 0}$ will then yield a new expression for g with $p'\ge p$, as in our previous case.

It remains to prove the claim. Since $p'<p$, the terms in g with $w_{}$-weight $p'$ must cancel out, i.e., $\sum _{j=1}^k {\text {in}}_{w_{}}(b_{j})\,{\text {in}}_{w_{}}(f_{j}) =0$. As in the proof of Lemma A.7, the fact that $({\text {in}}_{w_{}}(f_1), \ldots , {\text {in}}_{w_{}}(f_s))$ is a regular sequence in ${\mathcal {O}}$ ensures that

$$\begin{aligned} {\text {in}}_{w_{}}(b_{k}) = \sum _{j=1}^{k-1} c_j {\text {in}}_{w_{}}(f_{j}), \end{aligned}$$

where $c_j$ is either zero or a $w_{}$-homogeneous polynomial with $w_{}(c_j) = p'-w_{}(f_j)-w_{}(f_k)\ge 0$. It follows from here that the element $b_{k}':= b_{k} - \sum _{j=1}^{k-1} c_j \, f_j$ has weight $w_{}(b_{k}')> w_{}(b_{k})$, so $w_{}(b_{k}'\, f_{k}) > p'$.

An arithmetic manipulation allows us to rewrite g as follows:

$$\begin{aligned} g = \sum _{j=1}^{k-1} (b_j + c_j\, f_{k}) \,f_j + b_{k}'\,f_{k} + \sum _{j=k+1}^s b_j\, f_j \end{aligned}$$

(A.6)

By construction, the terms with minimum $w_{}$-weight only appear in the first of the three summands on the right-hand side of (A.6). Furthermore, the corresponding minimum weight $p''$ satisfies $p''\ge p'$ since $w_{}(b_jf_j)\ge p'$ for all j and $w_{}(c_j \, f_{k}f_j)\ge p'$ for $j<k$. This confirms the validity of our claim. $\square$

A standard commutative algebra result (see, e.g., [4, Lemma 10.23]) combined with Lemma A.8 yields:

Lemma A.9

Assume that the sequence $({\text {in}}_w(f_1),\ldots , {\text {in}}_w(f_s))$ is regular in ${\mathcal {O}}$. Then, the map $d_2:F \rightarrow R$ of filtered modules induces a surjection between their completions relative to the filtrations $(F_p)_{p \ge 0}$ and $(R_p)_{p \ge 0}$ respectively. More precisely, $\varprojlim F/F_p\twoheadrightarrow \varprojlim R/R_p$.

We let and be the ${\mathfrak {m}}$-adic completions of F and R respectively, which can be computed with standard methods. Indeed, by [4, Theorem 10.13], we have

$$\begin{aligned} {\hat{F}} \simeq F\otimes _{{\mathcal {O}}} {\hat{{\mathcal {O}}}}\quad \text { and }\quad {\hat{R}} \simeq R\otimes _{{\mathcal {O}}} {\hat{{\mathcal {O}}}}. \end{aligned}$$

(A.7)

The double inclusions in (A.2) allow us to compare the completions in Lemma A.9 induced by $(F_p)_{p \ge 0}$ and $(R_p)_{p\ge 0}$, with ${\hat{F}}$ and ${\hat{R}}$, respectively. More precisely,

Lemma A.10

Assume that the sequence $({\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_s))$ is regular in ${\mathcal {O}}$. Then, the completions appearing in Lemma A.9 agree with the ${\mathfrak {m}}$-adic ones, i.e.

$$\begin{aligned} \varprojlim F/F_p\simeq \varprojlim F/{\mathfrak {m}}^p F\simeq F\otimes _{{\mathcal {O}}} \hat{{\mathcal {O}}} \quad \text { and }\quad \varprojlim R/R_p\simeq \varprojlim R/{\mathfrak {m}}^p R \simeq R\otimes _{{\mathcal {O}}} \hat{{\mathcal {O}}}. \end{aligned}$$

(A.8)

Proof

We let $\ell :=\max \{w_{}(f_j): j=1,\ldots , s\}$. It suffices to prove the first isomorphism on each side of (A.8), since the remaining ones appear in (A.7). By (A.3), we have

$$\begin{aligned} E_p:=\bigoplus _{i} I_{p-w_{}(f_i)} \, e_{i}\qquad \text { and }\qquad F_p:=\bigoplus _{i<j} I_{p-w_{}(f_i)-w_{}(f_j)} \, e_{ij}\qquad \text { for each }p\ge 0. \end{aligned}$$

It follows from here that $I_p E \subseteq E_p \subseteq I_{p-\ell } E$ and $I_p F \subseteq F_p \subseteq I_{p-2\ell } F$ for each $p\ge 0$. Combining these inclusions with (A.2) yields:

$$\begin{aligned} E_{dp+\ell }{} & {} \subseteq I_{dp}E \subseteq {\mathfrak {m}}^p E\subseteq E_p\quad \text { and } \quad F_{dp+2\ell } \subseteq I_{dp} F \subseteq {\mathfrak {m}}^p F \subseteq F_p \nonumber \\{} & {} \qquad \text { for each }p\ge 0. \end{aligned}$$

(A.9)

The inclusions appearing on the right of (A.9) ensure that the filtrations $({\mathfrak {m}}^pF)_{p\ge 0}$ and $(F_p)_{p\ge 0}$ are cofinal in F. Thus, they yield isomorphic completions. This proves the first isomorphism in (A.8).

Next, consider the filtration $R_p$ from (A.4). First, notice that ${\mathfrak {m}}^pR \subseteq R_p$ by (A.9). To finish, we claim the existence of some $k\ge 0$ for which $R_{dp+(dk+\ell )} \subseteq {\mathfrak {m}}^pR$ for all $p\gg 0$. Indeed, by the Artin–Rees Lemma (see, e.g., [4, Theorem 10.10]), there exists an integer $k\ge 0$ satisfying

$$\begin{aligned} R\cap {\mathfrak {m}}^p E = {\mathfrak {m}}^{p-k}(R \cap {\mathfrak {m}}^k E) \qquad \text { for all } p\ge k. \end{aligned}$$

Therefore, combining this fact with property (A.9) we obtained the desired inclusion:

$$\begin{aligned} R_{dp+(dk+\ell )} = R \cap E_{d(p+k)+\ell } \subset R\cap {\mathfrak {m}}^{p+k} E = {\mathfrak {m}}^{p}(R\cap {\mathfrak {m}}^k E) \subset {\mathfrak {m}}^p R. \end{aligned}$$

We conclude that $({\mathfrak {m}}^pR)_{p\ge 0}$ and $(R_p)_{p\ge 0}$ are cofinal filtrations in R, so they yield isomorphic completions. $\square$

We let M be the cokernel of the map $d_2:F\rightarrow R$ given by (A.1), and we let be its ${\mathfrak {m}}$-adic completion. Lemma A.10 yields the following result:

Lemma A.11

Assume that the sequence $({\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_s))$ is regular in ${\mathcal {O}}$. Then, ${\hat{M}}=0$ and $M=0$. In particular, the Koszul complex (A.1) is exact at E.

Proof

By standard commutative algebra (see, e.g., [51, Corollaire 2, Chap. II.A.5]) we know that ${\hat{{\mathcal {O}}}}$ is a flat ${\mathcal {O}}$-module. Therefore, taking ${\mathfrak {m}}$-adic completion is an exact functor. Since ${\hat{F}}\rightarrow {\hat{R}}$ is surjective (by combining Lemmas A.9 and A.10) it follows that ${\hat{M}}=0$.

By [4, Theorem 10.17], the kernel of the canonical morphism $M\rightarrow {\hat{M}}$ is annihilated by an element of the form $(1+z)$ where $z\in {\mathfrak {m}}$. As ${\mathcal {O}}$ is a local ring, the element $(1+z)$ must be a unit of ${\mathcal {O}}$, thus $M=0$ as claim. The exactness of the Koszul complex at E follows immediately, as it is equivalent to the surjectivity of the morphism $d_2:F\rightarrow R$. $\square$

We end this appendix by proving its main result:

Proof of Theorem A.1

Since $({\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_s))$ is regular in ${\mathcal {O}}$, Lemma A.11 ensures that the Koszul complex (A.1) is exact at E. In turn, Proposition A.4 implies that $(f_1,\ldots , f_s)$ is a regular sequence in ${\mathcal {O}}$. This proves item (1) of the statement.

To finish, we must show that the $w_{}$-initial forms $\{{\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_s)\}$ generate the $w_{}$-initial ideal ${\text {in}}_{w_{}}(J){\mathcal {O}}$. By definition, the ideal generated by these forms is contained in ${\text {in}}_{w_{}}(J){\mathcal {O}}$. As ${\text {in}}_{w_{}}(J){\mathcal {O}}$ is generated over ${\mathcal {O}}$ by all elements ${\text {in}}_{w_{}}(g)$ with $g\in J$, the reverse inclusion will follow immediately if we show that ${\text {in}}_{w_{}}(g) \in ({\text {in}}_{w_{}}(f_1),\ldots , {\text {in}}_{w_{}}(f_s)) {\mathcal {O}}$. This identity is a direct consequence of Lemma A.8. Therefore, item (2) holds. This concludes our proof. $\square$