Eggers–Wall Tree in Plane Curve Singularities

• Eggers–Wall tree is a rooted decorated tree that represents the separation of branches and records characteristic exponents, denominator data, and contact orders.
• It reconstructs singularity invariants using methodologies based on Newton–Puiseux expansions, semigroups, key polynomials, and lotus constructions.
• The tree facilitates computational and valuative analyses, enabling recovery of intersection numbers and invariants even in positive characteristic settings.

The Eggers–Wall tree is a rooted decorated tree attached to a reduced plane curve singularity relative to a chosen smooth branch $L$. In its classical Newton–Puiseux form, the tree records how the Puiseux expansions of the branches separate from one another, together with numerical functions encoding characteristic exponents, denominator data, and contact orders. In more recent formulations, the same object is reconstructed from semigroups, key polynomials, or from a lotus associated to an embedded resolution, which makes the construction effective in settings where Puiseux series are unavailable or insufficient, notably in positive characteristic (Barroso et al., 2018, Barroso et al., 24 Feb 2025, Felipe et al., 28 Sep 2025).

1. Position in the theory of plane curve singularities

For a plane curve singularity, the combinatorial type may be encoded in several equivalent ways. The standard trio consists of the Eggers–Wall tree, the Enriques diagram, and the weighted dual graph of an embedded resolution. The 2019 survey emphasizes that these three trees contain the same information and that a lotus provides a geometric framework in which they all embed naturally (Barroso et al., 2019).

The construction is always relative to a smooth branch $L$ through the singular point. In the Newton–Puiseux description, one fixes local coordinates $(x,y)$ with $L=Z(x)$. The Eggers–Wall tree is then a rooted compact $\mathbb R$-tree whose root is labeled by $L$, whose leaves are labeled by the branches of the curve, and whose additional marked points record the characteristic exponents and mutual contacts of those branches. In the language of the survey, it may be viewed as a Galois quotient of the Kuo–Lu tree; in the valuative interpretation, it becomes a finite subtree of Favre–Jonsson’s valuative tree (Barroso et al., 2018, Barroso et al., 2019).

2. Classical Newton–Puiseux construction

Let $C$ be a reduced curve germ and $A\neq L$ an irreducible branch. Writing the Weierstrass polynomial of $A$ in coordinates with $L=Z(x)$, the Newton–Puiseux theorem produces its roots as fractional power series. The characteristic exponents of $L$0 relative to $L$1 are the numbers

$L$2

where $L$3 are distinct Newton–Puiseux roots of the defining polynomial of $L$4. They form a finite increasing sequence

$L$5

The Eggers–Wall segment $L$6 is a compact oriented segment with small end labeled by $L$7, large end labeled by $L$8, an exponent function

$L$9

and marked points precisely at the characteristic exponents. Its index function

$(x,y)$0

is defined as the index of the subgroup generated by $(x,y)$1 and the characteristic exponents $(x,y)$2, equivalently as the least common denominator of the exponents in a Puiseux expansion that are $(x,y)$3 (Barroso et al., 2018).

For two distinct branches $(x,y)$4, the order of coincidence is

$(x,y)$5

where $(x,y)$6 range over Newton–Puiseux roots of $(x,y)$7 and $(x,y)$8. The global Eggers–Wall tree $(x,y)$9 is obtained by taking the disjoint union of the branch segments $L=Z(x)$0 and gluing their initial parts up to exponent $L=Z(x)$1, identifying points with the same exponent. The resulting rooted tree has root $L=Z(x)$2, leaves given by the branches of $L=Z(x)$3, and marked points consisting of the root, the leaves, the ramification points created by gluing, and the points where the index function jumps (Barroso et al., 2018).

3. Natural functions and the recovery of singularity data

The classical Eggers–Wall tree carries three natural functions: the exponent function $L=Z(x)$4, the index function $L=Z(x)$5, and the contact complexity function $L=Z(x)$6. On a branch segment, contact complexity is defined by

$L=Z(x)$7

or equivalently by the explicit piecewise-linear expression obtained by integrating $L=Z(x)$8 between consecutive marked points. This function is strictly increasing along each root-to-leaf segment (Barroso et al., 2018).

The key structural identity is the tripod formula. If $L=Z(x)$9 and $\mathbb R$0 are two branches distinct from $\mathbb R$1, and $\mathbb R$2 denotes the tripod determined by those three ends, then

$\mathbb R$3

Equivalently,

$\mathbb R$4

Thus the tree does not merely encode branchwise Puiseux data; it also recovers pairwise intersection numbers (Barroso et al., 2018, Barroso et al., 24 Feb 2025).

The 2025 lotus paper makes this recoverability explicit. From $\mathbb R$5 one can recover, for each branch, its characteristic exponents, its Puiseux pairs and semigroup, its multiplicity and the intersection number $\mathbb R$6, and, for each pair of branches, the order of coincidence $\mathbb R$7 and the intersection number $\mathbb R$8 via contact complexity. The same paper also observes that

$\mathbb R$9

defines an ultrametric on the set of branches distinct from $L$0, and that the Eggers–Wall tree is essentially the corresponding dendrogram (Barroso et al., 24 Feb 2025).

4. Valuative interpretation and change of observer

A canonical embedding of the Eggers–Wall tree into Favre–Jonsson’s valuative tree gives a conceptual reformulation of its numerical decorations. For a reduced curve $L$1 and smooth branch $L$2, the map

$L$3

is an increasing continuous embedding of rooted trees, where $L$4 is the normalization of the semivaluation space by the condition $L$5. Under this embedding, the Eggers–Wall functions become pullbacks of valuative invariants: $L$6 Here $L$7, $L$8, and $L$9 are the relative log-discrepancy, relative multiplicity, and relative self-interaction functions on the valuative tree (Barroso et al., 2018).

This perspective yields an inversion theorem. If $C$0 is a second smooth branch of $C$1, then the valuative embeddings of $C$2 and $C$3 identify them canonically, and the corresponding triples of functions are explicitly related. The result generalizes the classical Abhyankar–Zariski inversion theorem from one branch to arbitrary reduced curves containing two smooth branches (Barroso et al., 2018).

The same paper proves a global limit statement: the projectivized valuative tree $C$4 is the projective limit of Eggers–Wall trees over all choices of reduced curves $C$5. In that sense, individual Eggers–Wall trees are finite approximants of a universal valuative object rather than isolated combinatorial gadgets (Barroso et al., 2018).

5. Lotus constructions and computational reconstruction

The lotus framework recasts Eggers–Wall trees in birational and combinatorial terms. A lotus is a finite contractible simplicial complex built from an active constellation of crosses associated to an embedded resolution. In this setting, the lateral boundary of the lotus is homeomorphic to the Eggers–Wall tree of a suitable completion $C$6 of the curve. Marked interior points of the tree correspond to rupture vertices of the lotus, and if a marked point $C$7 corresponds to a rupture vertex $C$8, then

$C$9

On each membrane, the index function is constant and equal to $A\neq L$0 along the terminal segment. With these assignments, the lateral boundary $A\neq L$1 becomes exactly the Eggers–Wall tree $A\neq L$2 (Barroso et al., 24 Feb 2025).

The construction is reversible. For a complete Eggers–Wall tree, one may choose a trunk decomposition, associate an abstract lotus $A\neq L$3 to each trunk via renormalized exponent data, and glue these abstract lotuses along equally labeled marked vertices. Applying the lotus-to-tree reconstruction to the resulting complex returns the original complete Eggers–Wall tree (Barroso et al., 24 Feb 2025).

This makes the Eggers–Wall tree part of a broader computational architecture. On the lotus one can compute log-discrepancies, orders of vanishing, multiplicities of strict transforms at infinitely near points, the semigroup of a branch, the delta invariant, and, in characteristic zero, the Milnor number. The 2019 survey states the same unification in more geometric language: the Eggers–Wall tree, Enriques diagram, and weighted dual graph all embed simultaneously inside the lotus, and the geometry of the lotus captures their numerical decorations (Barroso et al., 24 Feb 2025, Barroso et al., 2019).

6. Positive characteristic and the polar-curve problem

Positive characteristic forces substantial modifications of the classical picture. Puiseux series may fail to exist or may fail to parametrize all branches adequately; irreducible Weierstrass polynomials may have no Puiseux roots in $A\neq L$4. For this reason, recent work replaces the Newton–Puiseux definition by constructions based on blowups, semigroups of values, and key polynomials (Barroso et al., 24 Feb 2025, Felipe et al., 28 Sep 2025).

Two complementary normalizations now coexist in the literature. In the Newton–Puiseux convention, the exponent function is primary and contact complexity is obtained by integrating $A\neq L$5. In the semigroup/key-polynomial convention used for polar curves in arbitrary characteristic, an irreducible branch $A\neq L$6 determines a segment $A\neq L$7 with a primary contact complexity function

$A\neq L$8

marked points coming from key polynomials, a piecewise-constant index function $A\neq L$9, and an exponent function defined by

$A$0

For a reducible curve $A$1, the global tree is obtained by gluing the segments $A$2 along their initial parts using the logarithmic distance

$A$3

and the normalization is fixed by the tripod formula

$A$4

This gives an Eggers–Wall tree entirely in terms of semigroups, key polynomials, and intersection multiplicities (Felipe et al., 28 Sep 2025).

The most recent development concerns polar curves. For a plane curve germ $A$5 over an algebraically closed field of characteristic $A$6, the tree controls the factorization of the polar curve $A$7. Writing $A$8 for the product of the irreducible factors of $A$9 above a marked point $L=Z(x)$0, the paper defines the Eggers condition at $L=Z(x)$1 by

$L=Z(x)$2

Its main theorem states that

$L=Z(x)$3

In the irreducible case, this condition is equivalent to the arithmetic requirement that all multiplicity jumps $L=Z(x)$4 are prime to $L=Z(x)$5, which refines the earlier irreducible positive-characteristic results of García Barroso–Płoski. The same 2025 lotus paper adds that when all branches have Newton–Puiseux roots, the lotus-defined Eggers–Wall tree and the Newton–Puiseux Eggers–Wall tree have the same underlying tree and the same contact complexity, although their exponent and index functions may differ (Felipe et al., 28 Sep 2025, Barroso et al., 24 Feb 2025).

The modern picture therefore treats the Eggers–Wall tree simultaneously as a Puiseux-theoretic invariant, a valuative subtree, a boundary of a lotus, and an arithmetic detector for the behavior of polar curves. Its enduring role is to package the combinatorics of branch separation, intersection, and characteristic data into a form stable under translation between these different frameworks.