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Some lower bounds for the maximal number of A-singularities in algebraic surfaces. II

This paper extends recent constructions of algebraic surfaces with high numbers of A-type singularities to establish new lower bounds for the maximal number of such singularities in additional cases.

Original authors: Juan García Escudero

Published 2026-04-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan García Escudero

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect designing a very strange, multi-dimensional sculpture. This sculpture isn't made of stone or steel, but of pure math, floating in a complex, invisible space. Your goal? To make this sculpture as "bumpy" as possible.

In the world of algebraic geometry, these "bumps" are called singularities. Specifically, the paper focuses on a type of bump called an A-singularity (think of them as sharp points or cusps, like the tip of a star or the bottom of a valley).

The big question mathematicians have been asking for decades is: "For a sculpture of a certain size (degree $d$ ), what is the absolute maximum number of bumps we can pack onto it?"

We know the ceiling (the upper limit) for some sizes, but we are still trying to figure out the floor (the lower bound)—that is, "What is the guaranteed number of bumps we can definitely build?"

This paper, written by Juan García Escudero, is like a new instruction manual for building these bumpier sculptures. Here is how it works, explained through a few simple analogies:

1. The Recipe: Mixing Two Ingredients

To build these surfaces, the author uses a recipe that mixes two specific ingredients:

Ingredient A (The Landscape): A complex, multi-variable polynomial (let's call it $J$ ). Think of this as a rolling hilly terrain with specific peaks and valleys.
Ingredient B (The Controller): A simpler, one-variable polynomial (let's call it $B$ or the new $G$ ). Think of this as a dial or a knob that controls how the hills interact.

In a previous paper, the author showed how to mix these to get a certain number of bumps. In this paper, he says, "Let's tweak the 'Controller' dial." He introduces new versions of this dial (the polynomials $G$ ) that are slightly different but follow the same rules.

2. The Lego Blocks: Trees and Transformations

How do you design these "Controller" dials? The author uses a visual language involving trees.

Imagine a tree where the branches represent the shape of the polynomial.
The "leaves" and "trunks" of the tree determine where the bumps will appear.
The author has a set of transformation rules (like Lego instructions). You can take a tree and apply a rule (let's call them $\alpha$ and $\beta$ ) to add more branches or change the shape.

The Magic Trick:
By applying these rules in a specific sequence (like following a pattern: add a branch, then add another, then repeat), the author can generate an infinite series of new trees. Each new tree corresponds to a new polynomial, which in turn creates a new surface with more bumps than the last one.

It's like having a machine that takes a small, bumpy rock and, by pressing a button, turns it into a slightly larger rock with even more bumps, and you can press the button over and over again.

3. The Result: Packing More Bumps

The paper proves that by using these new "Lego instructions," we can construct surfaces with more A-singularities than we knew were possible before for certain sizes.

The Old Way: We knew we could build a surface with, say, 100 bumps.
The New Way: The author shows a method to build a surface of the same size with 105 bumps.
The Impact: This pushes the "lower bound" higher. It tells us, "Hey, we can definitely do at least this much!" It doesn't tell us the maximum possible (the ceiling), but it raises the floor, narrowing the gap between what we can do and what is theoretically possible.

4. Why Does This Matter?

You might ask, "Who cares about counting bumps on invisible math sculptures?"

The Puzzle: Mathematics loves to know the limits of what is possible. Knowing the exact limits helps us understand the fundamental structure of space and shape.
The Toolbox: By creating these new construction methods, the author provides a "toolbox" for other mathematicians. They can now use these specific tree-transformations to solve other problems or build even more complex shapes.
The Connection: The paper connects these abstract shapes to things like "planar trees" and "Belyi polynomials" (which are related to how numbers behave in deep ways). It shows that a problem about 3D bumps is actually deeply connected to the geometry of 2D trees.

Summary

Think of this paper as a new blueprint for a bumpy factory.
The author took an existing factory design, found a way to upgrade the assembly line (the new polynomials $G$ ), and proved that this new line can produce sculptures with more sharp points (singularities) than ever before. He didn't just find one new sculpture; he found a whole infinite family of them, proving that the universe of these mathematical shapes is even "bumpier" than we thought.

1. Problem Statement

The paper addresses a classical problem in algebraic geometry: determining the maximal number of singularities of a specific type ( $A_\nu$ ) that can exist on a complex algebraic surface of degree $d$ in the projective space $\mathbb{P}^3(\mathbb{C})$ .

While upper bounds for these maximal numbers are known for certain cases, the lower bounds (constructive existence of surfaces with high numbers of singularities) remain an active area of research. The author specifically aims to extend previous constructions (notably from a 2024 paper by the same author, referenced as [14]) to derive new lower bounds for maximal numbers of $A_\nu$ singularities for a broader series of cases, particularly for degrees $d = 3q$ .

2. Methodology

The core methodology relies on the construction of specific algebraic surfaces defined by affine equations of the form:
$J_d(u, v) + P(w) = 0$
where $J_d$ is a bivariate polynomial related to folding polynomials of the root lattice $A_2$ , and $P(w)$ is a carefully constructed univariate polynomial.

The construction process involves three main technical components:

A. Belyi Polynomials and Planar Trees

The author utilizes Belyi polynomials (univariate polynomials with at most two critical values, typically $\pm 1$ ). These polynomials are represented combinatorially by plane trees with bicoloring (black and white vertices).

Vertices: Represent critical points.
Degrees: The degree of a vertex corresponds to the multiplicity of the critical point plus one.
Critical Values: Black vertices correspond to critical value $-1$, and white vertices to critical value $+1$ .

B. Tree Transformations (The Language $L_E$ )

The paper introduces a formal language approach to generate new polynomials ( $G^{(t)}$ ) from initial polynomials ( $B^{(t)}$ ) found in previous work.

Initial Trees: The process starts with trees associated with polynomials $B^{(1)}, B^{(2)}, B^{(3)}$ defined in [14].
Transformations: A set of substitution rules (transformations $\alpha, \beta, \gamma, \delta, \bar{\delta}$ ) is applied to the trees. These transformations modify the tree structure (adding edges/vertices) while preserving specific arithmetic properties.
Condition (E): A triple $(d, N_{-1}, N_1)$ must satisfy a specific condition involving floor functions to ensure the resulting polynomial yields the desired singularity counts. The set of words (sequences of transformations) that maintain this condition forms the language $L_E$ .
Infinite Families: The author proves that for specific parameters (e.g., $\nu=2$ ), the language $L_E$ is infinite, allowing for the generation of an infinite series of polynomials with increasing degrees.

C. Surface Construction

The final surfaces are constructed by combining:

$J_d(u, v)$ : A polynomial with known critical points (nodes and cusps) derived from the $A_2$ root lattice.
$U^{(t)}_{d,\nu,\epsilon}(w)$ : A polynomial derived from the transformed trees $G^{(t)}$ , normalized to have critical values $0$ and $1$.

The singularities of the resulting surface $S_J$ are determined by the interaction of the critical points of $J_d$ and $U^{(t)}$ . Specifically, the number of $A_\nu$ singularities is calculated using the formula:
$N(S_J, A_\nu) = N_0(J, 1)N_{-1}(G, \nu) + N_{-1}(J, 1)N_1(G, \nu)$
where $N_0, N_{-1}$ denote the counts of critical points with specific values for the respective polynomials.

3. Key Contributions

Extension of Previous Constructions: The paper generalizes the construction from [14] by introducing new polynomials $G^{(t)}$ that replace the previous $B^{(t)}$ polynomials. This allows for the exploration of cases not previously covered.
Formal Language Framework: The introduction of the language $L_E$ and the systematic application of tree transformations ( $\alpha, \beta, \dots$ ) provides a rigorous combinatorial method to generate families of polynomials with controlled critical point structures.
New Lower Bounds: The paper derives explicit formulas for the number of $A_\nu$ $A_{ν}$ singularities for surfaces of degree $d=3q$ $d = 3 q$ .
- For $\nu > 2$ , the number of singularities is given by:
  $N(S_J, A_\nu) = \frac{d(d-1)}{2} \left\lfloor \frac{d}{\nu+1} \right\rfloor + \frac{d(d-3)}{3} + 1$
- For $\nu = 2$ (cusps), a specific formula involving a parameter $h$ (where $d=3(h+3)$ ) is provided.
Comparison with Folding Polynomials: The author demonstrates that the constructed surfaces $S_J$ possess strictly more singularities than surfaces constructed using folding polynomials ( $F A_2$ ) and standard Belyi polynomials ( $M_d$ ), specifically gaining additional cusps and $A_\nu$ singularities.

4. Key Results

Theorem 3.3: Establishes the exact count of $A_\nu$ $A_{ν}$ singularities for the constructed surfaces.
- For $\nu > 2$ , the count depends on the floor function $\lfloor \frac{d}{\nu+1} \rfloor$ , which relates to the number of critical points of the univariate component.
- For $\nu = 2$ , the count is a cubic polynomial in terms of the parameter $h$ .
Infinite Families: Lemma 2.3 and Lemma 2.4 prove that for various initial trees, the transformation sequences can be extended indefinitely, generating an infinite family of surfaces with increasing degrees and correspondingly high numbers of singularities.
Improvement over Existing Bounds: The paper shows that for $\nu > 1$ , the maximal number of singularities $\mu_{A_\nu}(d)$ satisfies:
$\mu_{A_\nu}(d) \geq N(S_J, A_\nu)$
where $N(S_J, A_\nu)$ is strictly greater than the counts obtained from previous constructions involving folding polynomials.

5. Significance

Advancement in Singularity Theory: This work pushes the boundaries of known lower bounds for $A$ -type singularities on algebraic surfaces. Since exact upper bounds are only known for very few cases, improving lower bounds is crucial for narrowing the gap between theoretical limits and constructive reality.
Combinatorial Algebraic Geometry: The paper effectively bridges combinatorial structures (plane trees, formal languages) with complex algebraic geometry. It demonstrates how manipulating tree topologies can directly control the analytic properties (singularities) of high-degree polynomials.
Explicit Constructions: Unlike non-constructive existence proofs, this paper provides explicit algebraic equations and transformation rules, allowing for the concrete generation of surfaces with these high singularity counts.
Generalization: By unifying previous results (such as those regarding cusps and other $A_\nu$ singularities in [10, 11]) under a single framework of tree transformations, the paper offers a more cohesive understanding of how to maximize singularities in algebraic surfaces.

In summary, García Escudero provides a robust combinatorial toolkit to generate algebraic surfaces with a maximized number of $A_\nu$ singularities, significantly improving the known lower bounds for degrees $d=3q$ and establishing a systematic method for future exploration in this domain.