2. Surface germs with unique $\mathcal {L}$-vector

In this section we prove parts (i) and (ii) of Theorem 1.1. More generally, we are interested in finding a suitable geometric condition yielding a class of complex surfaces $(X,0)$ whose $\mathcal{L}$-vector is completely determined by the topology of a resolution. To achieve this, we recall the precise definitions of the divisors $Z_{\rm max}(X,0)$ and $Z_{\rm min}$ that have been mentioned in the introduction, and determine a condition that guarantees their equality.

We begin by recalling the notion of Lipman cone. A more thorough discussion of the objects described in this section can be found in [Reference NémethiNém99]. Let $\Gamma$ be a finite connected graph without loops and such that each vertex $v\in V(\Gamma )$ is weighted by two integers $g(v)\geq 0$, called genus, and $e(v)$, called self-intersection. We assume that the incidence matrix induced by the self-intersections of the vertices of $\Gamma$, that is the matrix $I_\Gamma \in {{\mathbb {Z}}}^{V(\Gamma )}$ whose $(v,v')$th entry is $e(v)$ if $v=v'$, and the number of edges of $\Gamma$ connecting $v$ to $v'$ otherwise, is negative definite. Let $E=\bigcup _{v\in V(\Gamma )}E_v$ be a configuration of curves whose dual graph is $\Gamma$, so that $I_{\Gamma } = (E_v \cdot E_{v'})$, and consider the free additive group $\mathcal{G}$ generated by the irreducible components of $E$, that is,

\[ \mathcal{G} = \bigg\{D =\sum_{v \in V(\Gamma)} m_v E_v \biggm| m_v \in {{\mathbb{Z}}}\bigg\}. \]

By a slight abuse of notation, we refer to the elements of $\mathcal{G}$ as divisors on $\Gamma$. On $\mathcal{G}$ there is a natural intersection pairing $D\cdot D'$, described by the incidence matrix $I_{\Gamma }$, and a natural partial ordering given by setting $\sum m_v E_v \leq \sum m'_v E_v$ if an only if $m_v \leq m'_v$ for every $v \in V(\Gamma )$.

The Lipman cone of $\Gamma$ is the semi-group $\mathcal{E}^{+}$ of $\mathcal{G}$ defined as

\[ \mathcal{E}^{+} = \{D \in \mathcal{G} \,|\, D \cdot E_v \leq 0 \text{ for all } v \in V(\Gamma)\}. \]

A cardinal property of the Lipman cone $\mathcal{E}^{+}$, proven in [Reference ArtinArt66, Proposition 2], is that it has a unique nonzero minimal element $Z_{\rm min}$, called the fundamental cycle of $\Gamma$, and that moreover $Z_{\rm min}>0$, that is, the coefficients of $Z_{\rm min}$ are all positive. Observe that the existence of the fundamental cycle and the fact that $Z_{\rm min}>0$ are equivalent to the fact that $D>0$ for every nonzero divisor $D$ in $\mathcal{E}^{+}$.

Assume from now on that $\Gamma$ is the dual graph of a good resolution of a normal surface singularity $(X,0)$. Note that the Lipman cone, and therefore its fundamental cycle, only depend on the graph $\Gamma$, that is, on the topology of $(X,0)$, and not on the complex geometry of $(X,0)$; the fundamental cycle $Z_{\rm min}$ can be explicitly computed from $\Gamma$ by using Laufer's algorithm from [Reference LauferLau72, Proposition 4.1].

Consider now a germ of analytic function $f \colon (X, 0) \to ({{\mathbb {C}}}, 0)$. The total transform of $f$ by $\pi$ is the divisor $(f) = (f)_{\Gamma } + f^{*}$ on $X_\pi$, where $f^{*}$ is the strict transform of $f$ and $(f)_{\Gamma } = \sum _{v \in V(\Gamma )} m_v(f) E_v$ is the divisor supported on $E$ such that $m_v(f)$ is the multiplicity of $f \circ \pi$ along $E_v$. By [Reference LauferLau71, Theorem 2.6], we have

(1)\begin{equation} (f) \cdot E_v=0 \quad \text{for all}\ v \in V(\Gamma). \end{equation}

In particular, $(f)_\Gamma$ belongs to the Lipman cone $\mathcal{E}^{+}$ of $\Gamma$, and therefore the semi-group $\mathcal{A}_X^{+} = \{ (f)_{\Gamma } \,|\, f \in \mathcal{O}_{(X,0)}\}$ of $\mathcal{G}$ is contained in $\mathcal{E}^{+}$; it has a unique nonzero minimal element $Z_{\rm max}(X,0)$, which is called the maximal ideal divisor of $(X,0)$. Observe that the divisor $Z_{\rm max}(X,0)$ coincides with the cycle $(h)_{\Gamma }$ of a generic linear form $h \colon (X,0) \to ({{\mathbb {C}}},0)$, and that by the definition of the fundamental cycle we have $Z_{\rm min} \leq Z_{\rm max}(X,0)$.

The following proposition is the main result of this section.

Proof. Let $\pi ' \colon X_{\pi '} \to X$ be the minimal good resolution of $(X,0)$ which factors through the blowup of its maximal ideal and let $E_v$ be a component of $(\pi ')^{-1}(0)$. Let $\gamma ^{*}$ be a curvette of $E_v$, that is, a smooth complex curve germ intersecting transversely $E_v$ at a smooth point of $(\pi ')^{-1}(0)$, and let $h' \colon (X,0) \to ({{\mathbb {C}}},0)$ be a generic linear form of $(X,0)$ such that the strict transform of $(h')^{-1}(0)$ via $\pi '$ does not pass through the point $p=\gamma ^{*} \cap E_v$. Then the multiplicity $\operatorname {mult}(\gamma,0)$ of $\gamma =\pi '(\gamma ^{*})$ at $0$ can be computed as the intersection multiplicity of $\gamma$ with a Milnor fiber $\{h=t\}$ of $h$ in a small neighborhood of $0$. Let us choose local coordinates $(u,v)$ centered at $p$ such that $u=0$ is a local equation for $E_v$ and $v=0$ a local equation for $\gamma ^{*}$. Then by the definition of $m_v$ we have $(h' \circ \pi ) (u,v) = u^{m_v} \alpha (u,v)$ where $\alpha (u,v)$ is a unity in ${{\mathbb {C}}}\{u,v\}$ and, therefore, $m_v = \operatorname {mult}(\gamma,0)$.

If $v$ is an $\mathcal{L}$-node of $(X,0)$ and $h \colon (X,0) \to ({{\mathbb {C}}},0)$ is a generic linear form of $(X,0)$, so that $h^{-1}(0)$ is a generic hyperplane section of $(X,0)$, then there exists an irreducible component $\gamma$ of $h^{-1}(0)$ whose strict transform $\gamma ^{*}$ by $\pi '$ intersects $E_v$. By hypothesis, the curve $\gamma$ is smooth, therefore it has multiplicity $1$ and $\gamma ^{*}$ is a curvette of $E_v$. This proves that $m_v=1$.

Assume now that $\pi$ does not factor through the blowup of the maximal ideal, so that $\pi '= \pi \circ \alpha$, where $\alpha$ is a finite composition of point blowups. By minimality of $\pi '$ there exists an $\mathcal{L}$-node $v_0$ of $(X,0)$ which is associated with the exceptional component of one of the point blowups in $\alpha$. Let $\alpha _1$ be the first blowup in the sequence $\alpha$, that is, $\alpha _1$ is the blowup of $X_\pi$ at a point $p$ of $E_v \cap h^{*}$, where $E_v$ is a component of $\pi ^{-1}(0)$, and let $E_{w}$ be the exceptional curve of $\alpha _1$. As $h^{*}$ passes through $p$, we have $m_w=m_{w}(h) > m_v(h)\geq 1$. As this argument can be repeated for every blowup forming $\alpha$, we deduce that $m_{v_0}(h)>1$ as well, contradicting the first part of the proof. This implies that $\alpha$ must be an isomorphism, proving part (i).

To prove part (ii), write $Z_{\rm max}=Z_{\rm max}(X,0)=\sum _{v\in V(\Gamma )}m_vE_v$ and $Z_{\rm min}=\sum _{v\in V(\Gamma )}\widetilde m_vE_v$, and for every $v$ in $V(\Gamma )$ consider the non-negative integers

\[ l_v=-Z_{{\rm max}}\cdot E_v \quad \text{and}\quad {\tilde{l}_v}=-Z_{{\rm min}}\cdot E_v. \]

As $Z_{\rm min}\leq Z_{\rm max}$ by the definition of $Z_{\rm min}$, it is enough to prove that $Z_{\rm min}\geq Z_{\rm max}$. As $I_\Gamma$ is negative definite, it is therefore sufficient to show that the integer $(Z_{\rm min}-Z_{\rm max})\cdot E_v = l_v-{\tilde {l}_v}$ is at most zero for every vertex $v$ of $\Gamma$. Whenever $l_v = 0$, this follows immediately from the definition, so let us fix a vertex $v$ such that $l_v >0$. From part (i), we know that $m_v=1$. It follows from the inequality $0<\tilde {m}_v \leq m_v =1$ that $\tilde {m}_v =1$ as well. We therefore obtain

\[ l_v - {\tilde{l}}_v = (Z_{{\rm min}}-Z_{{\rm max}})\cdot E_v = \sum_{w\in V(\Gamma)} (\widetilde m_{w}-m_w)E_w\cdot E_v = \sum_{w\neq v}(\widetilde m_{w}-m_w)E_w\cdot E_v \leq 0, \]

because $E_w\cdot E_v \geq 0$ whenever $w\neq v$ and $\widetilde m_{w} \leq m_w$ at all vertices.

The hypothesis of Proposition 2.2 is quite weak, as it is satisfied by every normal surface germ with reduced tangent cone (in which case the components of a generic hyperplane section are not only smooth but also transverse, see, for example, [Reference Gonzalez-Sprinberg and Lejeune-JalabertGL97, § 1]), for example, by every minimal surface singularity. More generally, the hypothesis holds for all LNE surface germs, as was proven in [Reference Fernandes and SampaioFS19, Theorem 3.10]. In particular, the proposition implies parts (i) and (ii) of Theorem 1.1.

Observe that it follows by (1) that the vector $-I_{\Gamma _\pi }\cdot Z_{\rm max}(X,0)$ of ${{\mathbb {Z}}}^{V(\Gamma _\pi )}_{\geq 0}$ coincides with the $\mathcal{L}$-vector $L_\pi$ of $(X,0)$ considered in the introduction. Therefore, whenever $Z_{\rm max}(X,0)$ is determined by the topological type of $(X,0)$, the same holds true for $L_\pi$. We collect this result, which is the first step towards the proof of Corollary 1.3, in the following corollary.

3. A lemma on generic projections

In this section, we introduce three notions that prove fundamental in the remaining part of the paper, namely generic projections, non-archimedean links, and local degrees. We also prove an important result, Lemma 3.1, that shows the compatibility of generic projections with minimal resolutions.

We begin by discussing the notion of generic projection, which is based on seminal work of Teissier. Fix an embedding of $(X,0)$ in a smooth germ $({{\mathbb {C}}}^{n},0)$, and consider the morphism $\ell _{{\mathcal{D}}}\colon (X,0)\to ({{\mathbb {C}}}^{2},0)$ obtained as the restriction to $X$ of the projection along an $(n-2)$-dimensional linear subspace ${\mathcal{D}}$ of ${{\mathbb {C}}}^{n}$. Recall that whenever $\ell _{{\mathcal{D}}}$ is finite, the associated polar curve $\Pi _{{\mathcal{D}}}$ is the closure in $(X,0)$ of the ramification locus of the restriction of $\ell _{{\mathcal{D}}}$ to $X\smallsetminus \{0\}$, and the associated discriminant curve is the plane curve $\Delta _{\mathcal{C}}=\ell _{{\mathcal{D}}}(\Pi _{{\mathcal{D}}})$. The Grassmannian variety $\operatorname {Gr}(n-2,{{\mathbb {C}}}^{n})$ of $(n-2)$-planes in ${{\mathbb {C}}}^{n}$ contains an analytic dense open subset $\Omega$ such that, for every ${\mathcal{D}}$ in $\Omega$, the projection $\ell _{{\mathcal{D}}}$ is finite and the families $\{\Pi _{{\mathcal{D}}}\}_{{\mathcal{D}}\in \Omega }$ and $\{\Delta _{{\mathcal{D}}}\}_{{\mathcal{D}}\in \Omega }$ are both well behaved (for example, they are equisingular in a strong sense). We say that a morphism $\ell \colon (X,0)\to ({{\mathbb {C}}}^{2},0)$ is a generic projection of $(X,0)$ if $\ell =\ell _{{\mathcal{D}}}$ for some ${\mathcal{D}}$ in $\Omega$. A discussion of the properties satisfied by a generic projection, leading to a precise definition of $\Omega$, can be found in [Reference Neumann, Pedersen and PichonNPP20a, § 2], building on work of Teissier (see, in particular, [Reference TeissierTei82, Lemme-clé V 1.2.2]); we will come back to this matter later in this section.

We now recall the definition of the non-archimedean link $\operatorname {NL}(X,0)$ of the germ $(X,0)$. Indeed, our goal for this section is to study the map induced by a generic projection $\ell \colon (X,0)\to ({{\mathbb {C}}}^{2},0)$ on the dual graph of a good resolution of $(X,0)$. In principle, for this to make sense, it is necessary to choose a suitable good resolution $\pi \colon X_\pi \to X$ of $(X,0)$ and a compatible sequence of point blowups $\sigma \colon Y_\sigma \to {{\mathbb {C}}}^{2}$ of $({{\mathbb {C}}}^{2},0)$ in order for $\ell$ to induce a map $|\Gamma _\pi |\to |\Gamma _\sigma |$ between the topological spaces underlying $\Gamma _\pi$ and $\Gamma _\sigma$. In this paper, we use $\operatorname {NL}(X,0)$ as a convenient way of encoding intrinsically all the dual graphs of good resolutions of $(X,0)$; for this purpose, we can adopt the following ad hoc definition. Recall that, if $\pi \colon X_\pi \to X$ and $\pi ' \colon X_{\pi '} \to X$ are two good resolutions of $(X,0)$ such that $\pi '$ dominates $\pi$ (that is, $\pi '$ factors through $\pi$), then we have a natural inclusion $|\Gamma _\pi |\hookrightarrow |\Gamma _{\pi '}|$ between the topological spaces underlying the dual graphs $\Gamma _\pi$ and $\Gamma _{\pi '}$, and a retraction $|\Gamma _{\pi '}|\to |\Gamma _\pi |$ obtained by contracting the trees in $|\Gamma _{\pi '}|\smallsetminus |\Gamma _{\pi }|$. The non-archimedean link can then be seen as the inverse limit $\operatorname {NL}(X,0)=\varprojlim _{\pi }|\Gamma _\pi |$ in the category of topological spaces and with respect to the various retraction morphisms, where the limit runs over the poset of good resolutions of $(X,0)$, ordered by domination. In particular, $\operatorname {NL}(X,0)$ contains a copy the dual graph of each good resolution of $(X,0)$, and it can be seen as a compactification of the infinite union $\bigcup _\pi |\Gamma _\pi |$ of all the dual graphs of the good resolutions of $(X,0)$. As such, it can be thought of as a universal dual graph of the singularity $(X,0)$. To unburden the notation, in the remaining part of the paper we usually identify a dual graph $\Gamma _\pi$ with its image $|\Gamma _\pi |$ in $\operatorname {NL}(X,0)$. This point of view makes it convenient to think of $\mathcal{L}$- and $\mathcal{P}$-nodes abstractly as points of $\operatorname {NL}(X,0)$.

Traditionally, the non-archimedean link $\operatorname {NL}(X,0)$ is built as a space of normalized semivaluations on the complete local ring $\widehat {\mathcal{O}_{X,0}}$ of $X$ at $0$. In particular, if $\pi \colon X_\pi \to X$ is a good resolution of $(X,0)$ and $E_v$ is a component of its exceptional divisor $\pi ^{-1}(0)$, the corresponding vertex of $\Gamma _\pi$ is identified with the corresponding divisorial valuation $v\colon \widehat {\mathcal{O}_{X,0}}\to {{\mathbb {R}}}_+\cup \{+\infty \}$ defined by $v(f)=\operatorname {ord}_{E_v}(\pi ^{*}f)/m_v$, where $\operatorname {ord}_{E_v}(\pi ^{*}f)$ denotes the order of vanishing along $E_v$ of the pullback of $f$ via $\pi$. Throughout the paper, we freely make use of this terminology, calling divisorial point of $\operatorname {NL}(X,0)$ (or of a given dual graph $\Gamma _\pi$) any point that can arise in this way, and denoting by $E_v$ any exceptional curve corresponding to a divisorial point $v$. Observe that the subset of $\operatorname {NL}(X,0)$ consisting of its divisorial points is dense in the non-archimedean link; this corresponds to the fact that any given dual graph $\Gamma _\pi$ can be refined ad infinitum by passing to resolutions dominating $\pi$, subdividing each edge $e=[v,v']$ into smaller edges by successively blowing up double points starting with the blowup of $X_\pi$ at the point of $E_v\cap {E_{v'}}$ that corresponds to $e$. In particular, a divisorial point of $\operatorname {NL}(X,0)$ is contained in the interior of $e$ if and only if it is associated with an exceptional component that appears after blowing up only double points as above. We refer the reader to [Reference Belotto da Silva, Fantini and PichonBFP22, § 2.1] and [Reference FantiniFan18] for further details on this point of view.

The morphism $\ell \colon (X,0)\to ({{\mathbb {C}}}^{2},0)$ induces a natural map $\widetilde \ell \colon \operatorname {NL}(X,0)\to \operatorname {NL}({{\mathbb {C}}}^{2},0)$. From the point of view of semivaluations, this is simply defined functorially by pre-composing a semivaluation on $\widehat {\mathcal{O}_{X,0}}$ with the morphism of complete local rings $\widehat {\mathcal{O}_{{{\mathbb {C}}}^{2},0}} \to \widehat {\mathcal{O}_{X,0}}$ induced by $\ell$.

Concretely, $\widetilde \ell (v)$ can also be computed explicitly on a divisorial point $v$ of $\operatorname {NL}(X,0)$ as follows: we can find a sequence of point blowups $\sigma _{\ell,v}\colon Y_{\sigma _{\ell,v}} \to {{\mathbb {C}}}^{2}$ of $({{\mathbb {C}}}^{2},0)$ and a good resolution $\pi _{\ell,v}\colon X_{\pi _{\ell,v}}\to X$ of $(X,0)$ such that $v$ corresponds to a component $E_v$ of the exceptional divisor of $\pi _{\ell,v}$, the composition $\ell \circ \pi _{\ell,v} \colon X_{\pi _{\ell,v}} \to {{\mathbb {C}}}^{2}$ factors through a map $\widehat \ell \colon X_{\pi _{\ell,v}} \to Y_{\sigma _{\ell,v}}$ making the following diagram commute

(2)

and such that $E_v$ is mapped by $\widehat \ell$ surjectively onto a component $E_w$ of the exceptional divisor of $\sigma _{\ell,v}$; we then have $\widetilde \ell (v)=w$.

Let $\pi \colon X_\pi \to X$ be a good resolution of $(X,0)$. As $\ell$ is a finite map ramified precisely over the associated polar curve, the induced map $\widetilde \ell |_{\Gamma _\pi }\colon \Gamma _\pi \to \widetilde \ell ({\Gamma _\pi })$ is itself a finite cover, which on a set contained in the set of $\mathcal{P}$-nodes of $(X,0)$ that are contained in $\Gamma _\pi$ (not necessarily as vertices but possibly in the interior of some edges). In particular, $\widetilde \ell$ cannot contract an edge of $\Gamma _\pi$, but it may fold one if it contains a $\mathcal{P}$-node in its interior.

Observe that the map $\widetilde \ell$ clearly depends on the choice of $\ell$. Indeed, if $\ell '\colon (X,0)\to ({{\mathbb {C}}}^{2},0)$ is another generic projection obtained by composing $\ell$ with an automorphism $\phi$ of $({{\mathbb {C}}}^{2},0)$, then $\phi$ induces a nontrivial automorphism $\widetilde \varphi$ of $\operatorname {NL}({{\mathbb {C}}}^{2},0)$, and we have $\widetilde \ell = \widetilde \varphi \circ \widetilde \ell '$. Although, in general, two generic projections of $(X,0)$ do not differ by an automorphism of $({{\mathbb {C}}}^{2},0)$, it it possible to control this phenomenon if we restrict $\widetilde \ell$ to the dual graph $\Gamma _\pi$ of the minimal good resolution $\pi \colon X_\pi \to X$ of $(X,0)$ that factors through its Nash transform, as we explain in Lemma 3.1.

To do this, we need to dive deeper into the definition of generic projections, to be able to study the polar curves and the discriminant curves of $(X,0)$ in families. Let us begin by recalling the precise notion of strong equiresolution of singularities given in [Reference TeissierTei76, 3.1.1 and 3.1.5]. Given a morphism $\beta \colon M \to \Lambda$ with reduced fibers between smooth connected complex manifolds and a simple normal crossing divisor $E$ of $M$, we say that $\beta$ is simple (with respect to $E$) if $\beta$ is smooth and its restriction $\beta |_E \colon E \to \Lambda$ to $E$ is proper and locally a trivial deformation along its fibers. If we have another morphism $\sigma \colon M' \to M$, we say that $\sigma$ is $\beta$-compatible if the composition $\beta '=\beta \circ \sigma$ is simple (with respect to $E'=\sigma ^{-1}(E)$). Finally, given a (singular) subvariety $X$ of $M$, we say that an embedded resolution of singularities $\pi \colon \tilde {M} \to M$ of $X$ is a strong equiresolution (along $\Lambda$) of $X$ if $\pi$ is $\beta$-compatible and all of its restrictions $\pi _\lambda$ over $\lambda \in \Lambda$ are good embedded resolutions of $X_\lambda$.

According to [Reference TeissierTei82, Lemme-clé V 1.2.2] (see [Reference Neumann, Pedersen and PichonNPP20a, Proposition 2.3] for an English presentation), there exists an analytic open dense subset $\Omega$ of the Grassmannian $\operatorname {Gr}(n-2,{{\mathbb {C}}}^{n})$ where the family $\{(\Delta _{{\mathcal{D}}},{\mathcal{D}})\}_{{\mathcal{D}}\in \Omega }$ of discriminant curves, which can be seen as a surface in $(\mathbb {C}^{2},0)\times \Omega$ fibered over $\Omega$ via the projection $\beta \colon (\mathbb {C}^{2},0)\to \Omega$ on the second factor, admits a strong equiresolution

with $F=\sigma ^{-1}(\{0\}\times \Omega \})$ a simple normal crossing divisor of $\mathcal {Y}$.

For each ${\mathcal{D}}$ in $\Omega$, denote by $\sigma _{{\mathcal{D}}}\colon (\mathcal {Y}_{{\mathcal{D}}}, F_{{\mathcal{D}}})\to ({{\mathbb {C}}}^{2},0)$ the restriction of $\sigma$ to the fiber $\beta _{\mathcal{Y}}^{-1}({\mathcal{D}})$, which is a sequence of point blowups of $({{\mathbb {C}}}^{2},0)$. Given two elements ${\mathcal{D}}$ and ${\mathcal{D}}'$ of $\Omega$, this allows us to define an isomorphism of graphs $\eta _{{\mathcal{D}},{\mathcal{D}}'}\colon \Gamma _{\sigma _{{\mathcal{D}}}} \stackrel {\sim }{\longrightarrow } \Gamma _{\sigma _{{\mathcal{D}}'}}$ as follows. For each $v \in V(\Gamma _{\sigma _{{\mathcal{D}}}})$, if we denote by $F^{{\mathcal{D}}}_{v}$ the corresponding irreducible component of $\mathcal{F}=\sigma _{{\mathcal{D}}}^{-1}(0)$, there is a unique irreducible component $\mathcal{F}_v^{{\mathcal{D}}}$ of $\sigma ^{-1}(\{0\} \times \Omega )$ such that $F^{{\mathcal{D}}}_{v} = \mathcal{F}_v^{{\mathcal{D}}} \cap \sigma _{{\mathcal{D}}}^{-1}(0)$. We then set $\eta _{{\mathcal{D}},{\mathcal{D}}'}(v)=v'$, where $v'$ is the vertex of $\Gamma _{{\mathcal{D}}'}$ such that $\mathcal{F}_{v'}^{{\mathcal{D}}'}=\mathcal{F}_v^{{\mathcal{D}}}$ (that is, equivalently, such that $\mathcal{F}_v^{{\mathcal{D}}} \cap \sigma _{{\mathcal{D}}'}^{-1}(0)=F^{{\mathcal{D}}'}_{v'}$). This yields a bijection $V(\Gamma _{\sigma _{{\mathcal{D}}}}) \to V(\Gamma _{\sigma _{{\mathcal{D}}'}})$ which extends to a natural homeomorphism

\[ {\eta}_{{\mathcal{D}}, {\mathcal{D}}'} \colon \Gamma_{\sigma_{{\mathcal{D}}}} \to \Gamma_{\sigma_{{\mathcal{D}}'}} \]

defined on the divisorial points of $\Gamma _{\sigma _{{\mathcal{D}}}}$ as follows. Fix ${\mathcal{D}} \in \Omega$ and consider a divisorial point $v$ on an edge $[v_1,v_2]$ of $\Gamma _{\sigma _{{\mathcal{D}}}}$. Then $E^{{\mathcal{D}}}_v$ is created by a finite sequence of blowups of double points of the previous exceptional divisor, starting with the blowup of the point $F_{v_1}^{{\mathcal{D}}} \cap F_{v_2}^{{\mathcal{D}}}$. We can perform this blowups in family by blowing up along successive intersections of the form $\mathcal{F}_{w_1}^{{\mathcal{D}}} \cap \mathcal{F}_{w_2}^{{\mathcal{D}}}$, starting with the blowup along $\mathcal{F}_{v_1}^{{\mathcal{D}}} \cap \mathcal{F}_{v_2}^{{\mathcal{D}}}$. By composing this sequence of blowups with $\sigma$, we obtain a ($\beta$-compatible) morphism $\sigma _v \colon {\mathcal{Y}}_{\sigma _v} \to ({{\mathbb {C}}}^{2},0) \times \Omega$. The last blowup creates an irreducible new component $\mathcal{F}_v^{{\mathcal{D}}}$ in the exceptional divisor, and as before we define $v' = \eta _{{\mathcal{D}}, {\mathcal{D}}'}(v)$ by declaring that the corresponding irreducible component $F_{v'}^{{\mathcal{D}}'}$ is the intersection $\mathcal{F}_v^{{\mathcal{D}}} \cap ({\sigma '})^{-1}(0, {\mathcal{D}}')$. Observe that, because multiplicities are constant along a smooth family, we have $m_v=m_{\eta _{{\mathcal{D}},{\mathcal{D}}'}(v)}$ for every divisorial point $v$ of $\Gamma _{\sigma _{{\mathcal{D}}}}$.

The following lemma relating the graph $\Gamma _{\sigma _{{\mathcal{D}}}}$ to $\Gamma _{\pi }$ plays a crucial role in several arguments in the rest of the paper.

Lemma 3.1 Let $(X,0)$ be a normal surface singularity, let $\pi \colon X_\pi \to X$ be the minimal good resolution of $(X,0)$ that factors through the blowup of its maximal ideal and its Nash transform, and let $\Gamma _\pi \subset \operatorname {NL}(X,0)$ be the dual graph of $\pi$. Then for all ${\mathcal{D}}$ and ${\mathcal{D}}'$ in $\Omega$ the diagram

obtained by restricting to the graph $\Gamma _{\pi }$ the two induced morphisms of non-archimedean links $\widetilde \ell _{{\mathcal{D}}},\widetilde \ell _{{\mathcal{D}}'} \colon \operatorname {NL}(X,0) \to \operatorname {NL}({{\mathbb {C}}}^{2},0)$, is commutative.

Before moving to the proof of the lemma, which is rather technical, we observe that the homeomorphism ${\eta }_{{\mathcal{D}}, {\mathcal{D}}'} \colon \Gamma _{\sigma _{{\mathcal{D}}}} \to \Gamma _{\sigma _{{\mathcal{D}}'}}$ lifts naturally to an automorphism ${\eta }_{{\mathcal{D}}, {\mathcal{D}}'}$ of the dual graph $\Gamma _{\pi '}$ of any good resolution $\pi '\colon X_{\pi '}\to X$ of $(X,0)$. However, the commutativity ${\eta }_{{\mathcal{D}}, {\mathcal{D}}'} \circ \widetilde \ell _{{\mathcal{D}}}$ does not necessarily hold on the whole of $\Gamma _{\pi '}$. We defer an illustration of this phenomenon to Example 4.4, because showing this now would require a lengthy local computation, while after proving Lemma 4.1 we can give a more conceptual explanation.

As our needs go slightly beyond what was done by Teissier, let us explain how to adapt his constructions accordingly. We start by proving a technical lemma about resolution in families of surfaces, much in the spirit of [Reference TeissierTei76, 4.1 and 4.2].

Now, recall that we have an embedding of $(X,0)$ in $({{\mathbb {C}}}^{n},0)$ and let $\Phi \colon (X,0) \times \Omega \to ({{\mathbb {C}}}^{2},0) \times \Omega$ be the morphism defined by $\Phi (x,{\mathcal{D}}) = (\ell _{{\mathcal{D}}}(x), {\mathcal{D}})$, which is generically of maximal rank. Let $\pi \colon (X_{\pi },E) \to (X,0)$ be a good resolution of $(X,0)$ which factors through the blowup of its maximal ideal and through its Nash transform. We note that, by using [Reference LauferLau71, Lemma 5.2] (a special case of the direct image theorem of Grauert), resolution of singularities, and the universal property of blowups, there exists a sequence of blowups $\alpha \colon (\mathcal {Z},\mathcal{G}) \to (X_{\pi },E) \times \Omega$ and an analytic morphism $\Psi \colon (\mathcal {Z},\mathcal{G}) \to (\mathcal {Y},\mathcal{F})$ such that $\Psi ^{-1}(\mathcal{F})_{\mathrm {red}} = \mathcal{G}_{\mathrm {red}}$ and the following diagram is commutative

(3)

with $\beta _{\mathcal{Y}}=\beta \circ \sigma$ simple. Thanks to Lemma 3.2, up to shrinking the size of the open $\Omega$ if necessary, the morphism $\beta _{\mathcal{Z}}=\beta _{\mathcal{Y}}\circ \Psi$ is simple as well. We are now ready to complete the proof of Lemma 3.1.

Proof of Lemma 3.1 The map $\widetilde {\ell }|_{\Gamma _{\pi }}$ is determined by its restriction to the set of divisorial points of $\Gamma _{\pi }$, as those form a dense subset of $\Gamma _{\pi }$. As $\beta _{\mathcal{Z}}$ and $\beta _{\mathcal{Y}}$ are simple, for every pair of elements ${\mathcal{D}}, {\mathcal{D}}'$ of $\Omega$, the following diagram commutes

We now need to prove the result on the divisorial points of $\Gamma _{\pi }$ which are not vertices of $\Gamma _{\pi }$. It is sufficient to consider the case where $v$ is the divisorial point associated with the exceptional curve of the blowup $\pi '\colon (X_{\pi }',E')\to (X_{\pi },E)$ of center $E_{v_1} \cap E_{v_2}$, because the same argument can then be repeated verbatim for general sequence of point blowups. Observe that if $E_v \times \Omega$ is already a component of $\mathcal{G}$, then $\widetilde {\ell }_{{\mathcal{D}}} (v) \in \Gamma _{\sigma _{{\mathcal{D}}}}$ for every ${{\mathcal{D}}} \in \Omega$, and we conclude easily. If $\mathcal{G}_v = E_v \times \Omega$ is not a component of $\mathcal{G}$, we note that $\mathcal{G}_{v_1}\cap \mathcal{G}_{v_2} = (E_{v_1} \cap E_{v_2}) \times \Omega$ is an admissible center in $(\mathcal {Z},\mathcal{G})$, because all blowups in $\alpha$ are admissible. We therefore may perform this extra blowup $\alpha '\colon (\mathcal {Z}',\mathcal{G}') \to (\mathcal {Z},\mathcal{G})$, whose exceptional divisor $\mathcal{G}_v = E_v \times \Omega$ is trivial with respect to the family structure. Fix ${{\mathcal{D}}} \in \Omega$, set $w_1 = \ell _{{\mathcal{D}}}(v_1)$ and $w_2 = \ell _{{\mathcal{D}}}(v_2)$, and consider the associated components $\mathcal{F}_{w_1}^{{\mathcal{D}}}$ and $\mathcal{F}_{w_2}^{{\mathcal{D}}}$ of $\mathcal{F}$. Then, after performing a sequence of combinatorial blowups $\rho \colon (Y',\mathcal{F}') \to (Y,\mathcal{F})$, starting with blowing up the center $\mathcal{F}_{w_1}^{{\mathcal{D}}} \cap \mathcal{F}_{w_2}^{{\mathcal{D}}}$, the projection $\widetilde {\ell }_{{\mathcal{D}}'} (v)$ belongs to the graph of $\Gamma _{\rho _{{{\mathcal{D}}'}} \circ \sigma _{{\mathcal{D}}'}}$ for every ${\mathcal{D}}'$ in $\Omega$. We have obtained, without the need to shrink the size of $\Omega$, the following commutative diagram:

where $\beta _{\mathcal {Y}'}=\beta \circ \sigma \circ \rho$ and $\beta _{\mathcal {Z}'}=\beta _{\mathcal {Y}'}\circ \Psi '$ are simple morphisms. We conclude easily.

We conclude the section by recalling the definition of the local degree of a divisorial point $v$ of $\operatorname {NL}(X,0)$, as it will be very important in the remaining part of the paper. Let $\ell \colon (X,0)\to ({{\mathbb {C}}}^{2},0)$ be a generic projection of $(X,0)$ and consider the diagram (2). For each component $E_{\nu }$ of $\pi _{\ell,v}^{-1}(0)$ (respectively $E_{\nu '}$ of $\sigma _{\ell,v}^{-1}(0)$), let us choose a tubular neighborhood disc bundle $N(E_{\nu })$ (resp. $N(E_{\nu '})$), and consider the two sets

\[ \mathcal{N}(E_{v})=N(E_v)\smallsetminus \bigcup_{E_\nu\neq E_v}N(E_\nu)\quad \mbox{and}\quad \mathcal{N}(E_{\widetilde\ell(v)})=N(E_{\widetilde\ell(v)})\smallsetminus \bigcup_{E_{\nu'}\neq E_{\widetilde\ell(v)}}N(E_{\nu'}) \]

in $X_{\pi _{\ell,v}}$ and $Y_{\sigma _{\ell,v}}$, respectively. We can then adjust the disc bundles $N(E_{\nu })$ and $N(E_{\nu '})$ in such a way that the cover $\ell$ restricts to a cover

(4)\begin{equation} \ell_{v} \colon \pi_{\ell,v}(\mathcal{N}(E_{v})) \longrightarrow \sigma_{\ell,v}(\mathcal{N}(E_{\widetilde\ell(v)})) \end{equation}

branched precisely on the polar curve of $\ell$ (if $v$ is not a $\mathcal{P}$-node, the branching locus is just the origin). Using a resolution in family over $\Omega$ as in the proof of the Lemma 3.1, it is easy to deduce the following result.

Therefore, we can set $\deg (v) = \deg (\ell _{v})$. We call this integer the local degree of a generic projection of $(X,0)$ at $v$, or simply the local degree of $(X,0)$ at $v$. Note that if $\deg (v) = 1$, then the map (4) is an isomorphism.

4. Generic projections of LNE surfaces

In this section, we study LNE surface germs by establishing some properties related to their generic projections.

We begin by proving the invariance of multiplicities under generic projections, and showing a characterization of the $\mathcal{P}$-nodes of a LNE normal surface in terms of their local degrees. More precisely, we prove the following result.

Before delving into the proof of the lemma, let us recall the notions of inner and outer contact and that of inner rate. Let $(\gamma,0)$ and $(\gamma ',0)$ be two distinct real or complex curve germs on the surface germ $(X,0)\subset ({{\mathbb {C}}}^{n},0)$ and denote $S_{\epsilon }$ the sphere in ${{\mathbb {C}}}^{n}$ having center $0$ and radius $\epsilon >0$. The inner contact between $\gamma$ and $\gamma '$ is the rational number $q_{\operatorname {inn}}=q_{\operatorname {inn}}(\gamma, \gamma ')$ defined by

\[ d_{\operatorname{inn}} (\gamma \cap S_{\epsilon}, \gamma' \cap S_{\epsilon}) = \Theta(\epsilon^{q_{\operatorname{inn}}}), \]

where $\Theta$ stands for the big-Theta asymptotic notation of Bachmann–Landau, which is defined as follows: given two function germs $f,g\colon ([0,\infty ),0)\to ([0,\infty ),0)$ we say that $f$ is big-Theta of $g$, and we write $f(t) = \Theta (g(t))$, if there exist real numbers $\eta >0$ and $K >0$ such that ${K^{-1}}g(t) \leq f(t) \leq K g(t)$ for all $t\geq 0$ satisfying $f(t)\leq \eta$. The outer contact $q_{\operatorname {out}}(\gamma, \gamma ')$ is defined in an analogous way, by using the outer metric $d_{\operatorname {out}}$ instead of the inner metric $d_{\operatorname {inn}}$. Observe that if $(X,0)$ is LNE then $q_{\operatorname {inn}}(\gamma, \gamma ') = q_{\operatorname {out}}(\gamma, \gamma ')$. Recall that the inner rate $q_v$ of a divisorial point $v$ of $\operatorname {NL}(X,0)$ is defined as the inner contact $q_\mathrm {inn}(\gamma,\gamma ')$, where $\gamma,\gamma '\subset (X,0)$ are two curve germs that pullback to two curvettes through distinct points of the divisor $E_v$ associated with $v$ via any good resolution $\pi \colon X_\pi \to X$ of $(X,0)$ that makes the divisor $E_v$ appear. This definition only depends on the divisorial point $v$ (see [Reference Belotto da Silva, Fantini and PichonBFP22, Lemma 3.2]).

Proof. We begin by proving part (i). Write $\ell =\ell _{{\mathcal{D}}}$ and set $w = \widetilde {\ell }_{{\mathcal{D}}}(v)$. Consider maps

as in diagram (2) such that $\pi _{v,{\mathcal{D}}}$ is a good resolution of $(X,0)$ factoring through its Nash transform (and, therefore, through $\pi$), $\sigma _{v,{\mathcal{D}}}$ is a sequence of point blowups of $({{\mathbb {C}}}^{2},0)$ such that $w$ is associated with a component $E_w$ of $(\sigma _{v,{\mathcal{D}}})^{-1}(0)$, and the component $E_v$ of $(\pi _{v,{\mathcal{D}}})^{-1}(0)$ associated with $v$ is sent by $\widehat \ell$ surjectively onto $E_w$.

Take a curvette $\gamma ^{*}$ of $E_w$ which does not intersect a component of the strict transform of the discriminant curve $\Delta _{{\mathcal{D}}}$ of $\ell _{\mathcal{D}}$ and let $(\gamma,0)\subset ({{\mathbb {C}}}^{2},0)$ be the irreducible curve germ defined by $\gamma =\sigma _{v,{\mathcal{D}}}(\gamma ^{*})$, so that we have $m_w=\operatorname {mult}(\gamma )$. Up to replacing $\gamma ^{*}$ by a nearby curvette, among the components of $({\ell }_{{\mathcal{D}}})^{-1}(\gamma )$ we can find an irreducible curve germ $\widehat \gamma$ on $(X,0)$ whose strict transform by $\pi _{v,{\mathcal{D}}}$ is a curvette of $E_v$, so that we have $m_v=\operatorname {mult}(\widehat \gamma )$. We then have $\operatorname {mult}(\widehat {\gamma }) = k \operatorname {mult}(\gamma )$, where $k$ is the degree of the covering $\widehat {\gamma }\to \gamma$ induced by $\ell$.

We argue by contradiction. Assume that $\operatorname {mult}(\widehat \gamma )\neq \operatorname {mult}(\gamma )$, that is, that $k>1$. Our goal will be to construct two real arcs $\widehat \delta _1$ and $\widehat \delta _2$ inside $\widehat \gamma$ whose inner and outer contacts do not coincide; this will then imply that $(X,0)$ is not LNE, contradicting our hypothesis. To do so, we consider another generic projection $\ell _{{\mathcal{D}}'} \colon (X,0) \to (\mathbb {C}^{2},0)$, chosen to be generic with respect to the curve $\widehat {\gamma }$ as well, and set $\gamma '=\ell _{{\mathcal{D}}'}(\widehat {\gamma })$. Then the cover $\widehat {\gamma }\to \gamma '$ induced by $\ell _{{\mathcal{D}}'}$ has degree 1, and thus $\widehat \gamma$ and $\gamma '$ have the same multiplicity because $\operatorname {mult}(\widehat {\gamma }) = \mbox {degree}(\ell _{{\mathcal{D}}'}|_{\widehat {\gamma }})\operatorname {mult}(\gamma ')= \operatorname {mult}(\gamma ')$. Set $w' = \widetilde {\ell }_{{\mathcal{D}}'}(v)$. By Lemma 3.1, we have $w' = \eta _{{\mathcal{D}}, {\mathcal{D}}'}(w)$ and, thus, $m_{w'} = m_w$ and $q_{w'} = q_v = q_w$. Moreover, by the definition of $\eta _{{\mathcal{D}}, {\mathcal{D}}'}$, the strict transform of $\gamma '$ by $\sigma _{v,{\mathcal{D}}'}$ intersects $(\sigma _{v,{\mathcal{D}}'})^{-1}(0)$ in a smooth point $p$ of $E_{w'}$.

Observe that, because the plane curve germ $\gamma$ is the image through $\sigma _{v,{\mathcal{D}}}$ of a curvette of $E_w$, it has no characteristic Puiseux exponent strictly greater than the inner rate $q_w$ of $w$. On the other hand, the strict transform of $\gamma '$ by $\sigma _{v,{\mathcal{D}}'}$ cannot be a curvette of $E_{w'}$ because $\operatorname {mult}(\gamma ') = \operatorname {mult}(\widehat \gamma ) = k m_{w'}>m_{w'}$. Therefore, the minimal good embedded resolution of $\gamma '$ is obtained by composing $\sigma _{v,{\mathcal{D}}'}$ with a nontrivial sequence of point blowups, starting with the blowup of $Y_{\sigma _{v,{\mathcal{D}}'}}$ at $p$. Let $E_{w''}$ be the last irreducible curve created by this sequence, so that the strict transform of $\gamma '$ is a curvette of $E_{w''}$. Then the inner rate $q_{w''}$ of $E_{w''}$, which is strictly greater than $q_{w'}=q_w$, is a characteristic Puiseux exponent of $\gamma '$.

Let us choose an embedding $(X,0) \subset ({{\mathbb {C}}}^{n},0)$ and coordinates $(x_1,\ldots,x_n)$ of ${{\mathbb {C}}}^{n}$ such that $\ell _{{\mathcal{D}}'}(x) = (x_1,x_2)$ and $\gamma '$ is not tangent to the line $x_1=0$. Then, since $q_{w''}$ is a characteristic Puiseux exponent of $\gamma '$, we can find a pair of real arcs $\delta _1'$ and $\delta _2'$ among the components of the intersection $\gamma ' \cap \{x_1=t \mid t \in {{\mathbb {R}}}\}$ such that their contact $q(\delta _1',\delta _2')$ is equal to $q_{w''}$ (we refer to [Reference Neumann and PichonNP14, § 3] for details on this classical result about Puiseux expansions). Let $\widehat \delta _1$ and $\widehat \delta _2$ be two liftings of $\delta _1'$ and $\delta _2'$ via $\ell '$. As the projection $\ell _{{\mathcal{D}}'}$ is generic with respect to $\widehat \gamma$, it induces by [Reference TeissierTei82, pp. 352–354] a bi-Lipschitz homeomorphism for the outer metric from $\widehat {\gamma }$ onto $\gamma '$ and, therefore, the outer contacts $q_{\operatorname {out}}(\widehat {\delta }_1,\widehat {\delta }_2)$ and $q(\delta _1',\delta _2')$ coincide, so that, in particular, we have

(5)\begin{equation} q_{\operatorname{out}}(\widehat{\delta}_1,\widehat{\delta}_2)=q_{w''}>q_v. \end{equation}

We now show that the inner contact $q_{\operatorname {inn}}(\widehat {\delta }_1,\widehat {\delta }_2)$ between $\widehat {\delta }_1$ and $\widehat {\delta }_2$ is at most $q_v$, which will yield the contradiction we were after. Observe that the inner contact $q = q_{\operatorname {inn}}^{X}(\widehat {\delta }_1,\widehat {\delta }_2)$ between $\widehat {\delta }_1$ and $\widehat {\delta }_2$ can also be computed as $d^{F_t}_{\operatorname {inn}}(\widehat {\delta }_1(t),\widehat {\delta }_2(t)) = \Theta (t^{q})$, where $d^{F_t}_{\operatorname {inn}}(\widehat {\delta }_1(t),\widehat {\delta }_2(t))$ denotes the inner distance between $\widehat {\delta }_1(t)$ and $\widehat {\delta }_2(t)$ inside the Milnor fiber $F_t = X \cap \{x_1=t\}$, that is, the distance measured by taking the infimum of the inner lengths of the paths joining $\widehat {\delta }_1(t)$ to $\widehat {\delta }_2(t)$ inside $F_t$. This is a consequence of the fact that, by [Reference Birbrair, Neumann and PichonBNP14] and in the language therein, the subset $\pi ({\mathcal{N}}(E_v))$ of $(X,0)$ is a $B(q_v)$-piece fibered by the restriction of the generic linear form $x_1$ whenever $q_v>1$, whereas it is a conical piece if $q_v=1$.

To conclude, consider a small disc $D$ contained in the divisor $E_v$ and centered at the point $\widehat {\gamma }^{*}\cap E_v$ and let $N \cong D \times D'$ be a trivialization of the normal disc-bundle to $E_v$ over $D$ such that $\widehat {\gamma }^{*} = \{0\} \times D'$. The intersection $F_t \cap \pi (N)$ consists of $m_v$ disjoint discs each centered at one of the $m_v$ distinct points of $\widehat {\gamma } \cap F_t$. As $\delta _1(t)$ and $\delta _2(t)$ are two of these points, then they are the centers of two of these discs, $D_1$ and $D_2$, respectively. As these two discs have diameters $\Theta (t^{q_v})$, any path from $\widehat {\delta }_1(t)$ to $\widehat {\delta }_2(t)$ inside $F_t$ will have intersections with $D_1$ and $D_2$ of length at least $\Theta (t^{q_v})$. Therefore, $q_{\operatorname {inn}}( \widehat {\delta }_1,\widehat {\delta }_2) \leq q_v$, and so $q_{\operatorname {inn}}( \widehat {\delta }_1,\widehat {\delta }_2) < q_{\operatorname {out}}(\widehat {\delta }_1,\widehat {\delta }_2)$, which contradicts the fact that $(X,0)$ is LNE. This completes the proof that $m_v=m_{\widetilde \ell _{{\mathcal{D}}}(v)}$.

Let us now prove part (ii). If $v$ is a $\mathcal{P}$-node, then it immediately follows from the definition of degree that $\deg (v)>1$, because the cover $\ell$ is ramified in a neighborhood of the polar curve. Assume that $v$ is not a $\mathcal{P}$-node and that $\deg {v}>1$. We use again the plane curve $\gamma = \sigma _{v,{\mathcal{D}}}(\gamma *)$ introduced in the proof of part (i).

By the definition of $\deg (v)$, the curve $\ell ^{-1}(\gamma )$ has $k_v$ irreducible components whose strict transforms by $\pi _{v, {\mathcal{D}}}$ are curvettes of $E_v$, where $k_v$ divides $\deg (v)$, and we have $m_{v} = m_{\widetilde {\ell }(v)}{\deg (v)}/{k_v}$. As $m_{v} = m_{\widetilde {\ell }(v)}$ by part (i), then $\deg (v) = k_v$, so $k_v>1$. Let $\widehat {\gamma }_1$ and $\widehat {\gamma }_2$ be two components of $\ell ^{-1}(\gamma )$ whose strict transforms by $\pi _{v, {\mathcal{D}}}$ are curvettes of $E_v$ (as was the case in part (i), two such components can always be found after replacing $\gamma ^{*}$ by a nearby curvette if necessary), and let us consider two real arcs $\widehat {\delta }_1 \subset \widehat {\gamma }_1$ and $\widehat {\delta }_2 \subset \widehat {\gamma }_2$ such that $\ell _{{\mathcal{D}}}(\widehat {\delta }_1) = \ell _{{\mathcal{D}}}(\widehat {\delta }_2)$. By the definition of $q_v$, we have $q_{\operatorname {inn}}(\widehat {\gamma }_1, \widehat {\gamma }_2) = q_v$ and then $q_{\operatorname {inn}}(\widehat {\delta }_1, \widehat {\delta }_2) = q_v$. As $v$ is not a $\mathcal{P}$-node, the lifted Gauss map ${\lambda }$ on $X_\pi$ (see [Reference Neumann, Pedersen and PichonNPP20a, Definition 6.11]) is constant along $E_v$ and we then have $\lambda (p_1) = \lambda (p_2)$, see [Reference Neumann, Pedersen and PichonNPP20a, p. 19]. By [Reference Neumann, Pedersen and PichonNPP20a, Lemma 9.1], this implies that $q_{\operatorname {inn}}(\delta _1, \delta _2) < q_{\operatorname {out}}(\delta _1, \delta _2)$, therefore $(X,0)$ is not LNE, contradicting the hypothesis.

As a simple consequence of Lemma 4.1(ii) we deduce the following result.

Example 4.4 Let us show with an example that the minimality of $\pi$ is a necessary hypothesis in Lemma 4.1. Let $(X,0)$ be the standard singularity $A_2$, which is the hypersurface singularity in $({{\mathbb {C}}}^{3},0)$ defined by the equation $x^{2}+y^{2}+z^{3}=0$. A good resolution $\pi \colon X_\pi \to X$ of $(X,0)$ can be obtained by the method described in [Reference LauferLau71, Chapter II]. It considers the generic projection $\ell = \ell _{{\mathcal{D}}} = (y,z) \colon (X,0) \to ({{\mathbb {C}}}^{2},0)$ and, given a suitable embedded resolution $\sigma _{\Delta } \colon Y_{\sigma _{\Delta }} \to {{\mathbb {C}}}^{2}$ of the associated discriminant curve $\Delta \colon y^{2}+z^{3}=0$, gives a simple algorithm to compute a resolution of $(X,0)$ as a cover of $Y$. In this example, $\Delta$ is a cusp and the dual graph $\Gamma _{\sigma _\Delta }$ of its minimal embedded resolution $\sigma _\Delta$ is depicted on the left of Figure 1. Its vertices are labeled as $w_0$, $w_1$, and $w_2$ in their order of appearance as exceptional divisors of point blowups in the resolution process, the negative number attached to each vertex denotes the self-intersection of the corresponding exceptional curve, the positive numbers in parentheses denote the multiplicities, and the arrow denotes the strict transform of $\Delta$. In this case, Laufer's method gives us the dual graph of a good resolution $\pi _\ell$ of $(X,0)$ such that $\ell \circ \pi _\ell$ factors through $\sigma _\Delta$, appearing as the graph in the middle of Figure 1. Again, all exceptional components are rational, each vertex is decorated by the self-intersection of the corresponding exceptional curve and with its multiplicity, the arrow denotes the strict transform of $\Delta$, and the vertices are labeled in a way that $\widetilde \ell (v_0)=\widetilde \ell (v'_0)=w_0$, $\widetilde \ell (v_1)=w_1$, and $\widetilde \ell (v_2)=w_2$. Observe that the vertex $v_1$ has multiplicity 2, but it is sent by $\widetilde \ell$ to the vertex $w_1$, which has multiplicity 1. However, the rational curve $E_{v_1}$ associated with the vertex $v_1$ has self-intersection $-1$ and can, thus, be contracted. The resulting map $\pi \colon X_\pi \to X$, which no longer factors through $\sigma _\Delta$, is the minimal resolution of $(X,0)$ factoring through its Nash transform. Observe that its $\mathcal{P}$-node $v_2$ can also be contracted, yielding the minimal good resolution of $(X,0)$, which in this case does not factor through the Nash transform of $(X,0)$.

Figure 1. Dual resolution graphs for the plane curve $\Delta$ (left) and for the surface singularity $X=A_2$ (middle and right).

In the proof of Lemma 4.1, the minimality of $\pi$ is only required to apply Lemma 3.1. Therefore, this example also shows how the commutativity of the diagram of Lemma 3.1 may fail to hold on a larger dual graph such as $\Gamma _{\pi _\ell }$.

We can now move our focus to the morphism $\widetilde \ell$ induced by a generic projection $\ell \colon (X,0)\to ({{\mathbb {C}}}^{2},0)$, and more precisely to its restriction to the dual graph $\Gamma _\pi$ of some good resolution of $(X,0)$. Recall that, given a graph $\Gamma$, we denote by $V(\Gamma )$ the set of its vertices. In general, even whenever $\pi$ factors through the Nash transform of $(X,0)$, it is not possible to find a suitable sequence of point blowups $\sigma \colon Y \to {{\mathbb {C}}}^{2}$ of $({{\mathbb {C}}}^{2},0)$ such that $\widetilde \ell$ induces a morphism of graphs $\widetilde \ell |_{\Gamma _\pi }\to \Gamma _\sigma$, because to make the elements of $\widetilde \ell (V(\Gamma _\pi ))$ appear among the vertices of $\Gamma _\sigma$, one usually introduces too many additional vertices, so that the image $\widetilde \ell (e)$ of some edge $e$ of $\Gamma _\pi$ is not an edge of $\Gamma _\sigma$, but only a string of several edges. Remarkably, thanks to Lemma 4.1(ii), in the case of LNE surfaces we can control this phenomenon completely. Indeed, the following proposition explains that in this case we do obtain a morphism of graphs, provided that we restrict our attention to a subgraph of $\Gamma _\pi$ that does not contain a $\mathcal{P}$-node of $(X,0)$ in its interior.

Let $v$ be a vertex of $\Gamma _{\pi }$ which is not a $\mathcal{P}$-node, let $W$ be the connected component of $\Gamma _\pi \smallsetminus \{\mathcal{P}{\rm -nodes}\}$ containing $v$, and let $\Gamma _0$ be any subgraph of $\Gamma _\pi$ contained in the closure of $W$ and such that $v\in V(\Gamma _0)$ and all edges of $\Gamma _\pi$ at $v$ are edges of $\Gamma _0$. Then the last part of the statement of Proposition 4.5 tells us that $g(E_v)=g(E_{\widetilde \ell (v)})=0$ and $E_v^{2}=E_{\widetilde \ell (v)}^{2}$, where the self-intersection of $E_v$ is computed in $X_\pi$ and that of $E_{\widetilde \ell (v)}$ is computed in $Y_{\sigma _{\Gamma _0}}$.

Proof. Write $\partial W = \{z_1,\ldots,z_n\}\subset V(\Gamma _\pi )$. Observe that, because $\Gamma _0$ contains no $\mathcal{P}$-node in its interior, $\widetilde \ell$ does not fold it, and, thus, $\partial (\widetilde \ell (W)) = \widetilde \ell (\partial W)$ as subsets of $V(\Gamma _{\sigma _{\Gamma _0}})$. If $W$ contains at least one vertex of $\Gamma _0$, that is, if $\Gamma _0$ has at least a vertex which is not a point of $S$, denote by $\Gamma ^{\circ }_0$ the maximal subgraph of $\Gamma _0$ contained in $W$ and set $V(\Gamma ^{\circ }_0) = \{v_1, \ldots, v_r\}$, so that $V(\Gamma _0) = \{v_1, \ldots, v_r,z_1,\ldots,z_n\}$. Let $U$ be a tubular neighborhood of the curve $C= E_{v_1} \cup \cdots \cup E_{v_r}$ in $X_{\pi }$. As the incidence matrix of $\Gamma ^{\circ }_0$ is negative definite, the analytic contraction $\eta \colon (U,C) \to (S,p)$ of the curve $C$ onto a point $p$ defines a normal surface singularity $(S,p)$. Observe that, because $\pi$ is the minimal resolution of $(X,0)$ which factors through its Nash transform, the only components of $\pi ^{-1}(0)$ that could be contracted while retaining smoothness of the ambient surface are associated with $\mathcal{P}$-nodes of $(X,0)$. As $\Gamma ^{\circ }_0\subset W \subset \Gamma _{\pi }\smallsetminus S$ contains no $\mathcal{P}$-node, this implies that $\eta \colon (U,C) \to (S,p)$ is the minimal good resolution of the surface germ $(S,p)$. On the other hand, if $W$ contains no vertex of $\Gamma _0$, then $\Gamma _0$ consists of two vertices $v$ and $v'$ in $S$ and a single edge corresponding to an intersection point $p=E_v \cap E_{v'}$, in which case we set $(U,C) = (S,p) = (X_\pi,p)$ and $\eta = \mathrm {Id}_{U}$.

Let $\widehat \pi \colon X_{\widehat \pi } \to X$ be the minimal resolution of $(X,0)$ that factors through its Nash transform and such that $\ell \circ \widehat \pi$ factors through $\sigma _{\Gamma _0}$ via a map $\widehat {\ell }\colon X_{\widehat \pi }\to Y_{\Gamma _0}$. Then $\widehat \pi$ factors through $\pi$ by minimality of the latter, so that we obtain a commutative diagram as follows.

Set $\widehat {U}=\beta ^{-1}(U)$ and $\widehat C = \beta ^{-1}(C)$, so that $(\widehat U,\widehat C)$ contracts to $(S,p)$ via $\widehat \pi = \pi \circ \beta$. Set $U' = \widehat {\ell }(\widehat {U})$ and consider the curve $C' = \widehat \ell (\widehat C) = E_{w_1} \cup \cdots \cup E_{w_s} \subset \sigma _{\Gamma _0}^{-1}(0)$ in $Y_{\Gamma _0}$. Observe that, because $\partial (\widetilde \ell (W)) = \widetilde \ell (\partial W) \subset V(\Gamma _{\sigma _{\Gamma _0}})$, we have $\{w_1, \ldots, w_s\}=\widetilde {\ell }(W) \cap V(\Gamma _{\sigma _{\Gamma _0}})$. Moreover, because $\widehat U \cap \widehat \pi ^{-1}(0) \subset \widehat C \cup E_{z_1} \cup \cdots \cup E_{z_n}$ and no curve among the $E_{z_i}$ is contracted by $\widehat \ell$, we deduce that $U'$ is open in $Y_{\Gamma _0}$. It follows that $U'$ is a tubular neighborhood of $C'$. This also shows that the set $\widetilde \ell (\Gamma _0)$ is the closure of the connected component of $\Gamma _{\sigma _{\Gamma _0}}\smallsetminus \widetilde {\ell }(\partial W)$ which contains $\widetilde {\ell }(W)$. To establish the proposition, it is then sufficient to prove that $\ell$ induces an isomorphism between the pairs $(U,C)$ and $(U',C')$.

Similarly as above, the contraction of the curve $C'$ in $U'$ defines a normal surface singularity $(S',p')$ and an analytic map $\eta ' \colon (U',C') \to (S',p')$ which is a good resolution of $(S',p')$. Moreover, the restriction $\widehat {\ell }|_{U}$ induces a finite analytic map $\check{\ell } \colon (S,p)\to (S',p')$. As $W$ contains no $\mathcal{P}$-node, by Lemma 4.1 we have $\deg (w) = 1$ for every divisorial point $w$ of $W \cap \Gamma _0$. This implies that $\widehat {\ell }$ is a one-sheeted analytic covering between normal complex analytic spaces, thus an isomorphism by [Reference RemmertRem94, Proposition 14.7]. It follows that $\check{\ell }^{-1} \circ \eta '$ is a good resolution of $(S,p)$. Therefore, by minimality of the resolution $\eta$, the map $\check{\ell }^{-1} \circ \eta '$ factors through $\eta$ via a finite sequence of point blowups $\alpha \colon (U',C') \to (U,C)$, so that we obtain the following commutative diagram.

It remains to show that $\alpha$ is an isomorphism. If this is not the case, then the exceptional component of the last point blowup forming $\alpha$, which is contractible by definition, is the image through $\widehat \ell$ of the exceptional component of one of the point blowups forming $\beta$. As this contradicts the minimality condition in the definition of $\sigma _{\Gamma _0}$, this proves both parts (ii) and (iii) of the proposition. Part (i) is then a consequence of the minimality of $\pi$.

What might prevent Proposition 4.5 from holding globally on $\Gamma _\pi$ is that, for example, there might exist an edge $e$ of $\Gamma _\pi$ such that $\widetilde \ell (e)$ contains in its interior one (and, for the sake of the example, exactly one) point of the form $\widetilde \ell (v)$ for some vertex $v$ elsewhere in $\Gamma _\pi$. However, whenever this happens it is always possible to refine the graph $\Gamma _\pi$, performing a blowup of the double point of the exceptional divisor of $X_\pi$ that corresponds to $e$ and, thus, subdividing the edge $e$ by adding a new vertex $w$, and this vertex satisfies $\widetilde \ell (w)=\widetilde \ell (v)$. Observe that, if $(X,0)$ were arbitrary, this could still fail to give a morphism of graphs because the vertex associated with the blowup would not necessarily be sent to $\widetilde \ell (v)$ by $\widetilde \ell$. The fact that this does not occur in the case of LNE surfaces and that, therefore, we can refine $\Gamma _\pi$ to obtain a morphism of graphs, is the content of the following corollary.

Observe that the resulting morphism of graphs $\widetilde \ell |_{\Gamma _{\pi '}} \colon \Gamma _{\pi '} \to \Gamma _{\sigma _\ell }$ is not surjective, as is clear from Example 4.4. This issue is discussed further in § 7.

We conclude the section by discussing a remarkable property of the Nash transforms of LNE normal surface germs. The singularity $(S,p)$ appearing in the course of the proof of Proposition 4.5, being isomorphic to the singularity $(S',p')$ appearing in a modification of $({{\mathbb {C}}}^{2},0)$, is sandwiched, which means that it admits a proper bimeromorphic morphism to a smooth surface germ. Sandwiched singularities play an important role in Spivakovsky's proof of resolution of singularities of surfaces via normalized Nash transforms [Reference SpivakovskySpi90], because Hironaka [Reference HironakaHir83] proved that it is possible to reduce any singularity to a sandwiched singularity by a finite sequence of normalized Nash transforms. As Proposition 4.5 applies, in particular, to any connected component of the complement in $\operatorname {NL}(X,0)$ of its $\mathcal{P}$-nodes, and the $\mathcal{P}$-nodes are precisely the divisorial valuations corresponding to the exceptional components of the Nash transform of $(X,0)$, we deduce the following result.

5. Inner rates on LNE surface germs

In this section, we move to the study of the inner rates of LNE surface germs, whose definition was recalled immediately before the proof of Lemma 4.1, and prove parts (iii) and (iv) of Theorem 1.1.

We begin by endowing the dual graph of a good resolution of $(X,0)$ with a natural metric. Let $\pi \colon X_\pi \to X$ be a good resolution of $(X,0)$ factoring through the blowup of its maximal ideal, and denote by $|\Gamma _\pi |$ the topological space underlying the graph $\Gamma _\pi$. We endow $|\Gamma _\pi |$ with the metric defined by declaring that the length of an edge connecting two vertices $v$ and $w$ is equal to $1/\operatorname {lcm}(m_v,m_w)$, and denote by $d$ the associated distance function. Observe that, because the exceptional component of the blowup of an intersection point between the two components associated with $v$ and $w$ has multiplicity $m_v+m_w$, and $1/\operatorname {lcm}(m_v,m_w)=1/\operatorname {lcm}(m_v,m_v+m_w)+1/\operatorname {lcm}(m_v+m_w,m_w)$, the metric on $|\Gamma _\pi |$ is compatible with subdividing the edges of the graph $\Gamma _\pi$ by blowing up $X_\pi$ at double points of $\pi ^{-1}(0)$, and thus induces a metric on $\operatorname {NL}(X,0)$. The reader should be warned that this metric on $\Gamma _\pi$ is not the same as that defined in [Reference Belotto da Silva, Fantini and PichonBFP22, § 2.1], albeit it is strictly related to the latter and was already briefly used in Lemma 5.5 of [Reference Belotto da Silva, Fantini and PichonBFP22].

The following proposition is strictly stronger than part (iii) of Theorem 1.1, as it computes inner rates on the whole $\operatorname {NL}(X,0)$ rather than on a specific resolution graph.