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

Juan García Escudero

TL;DR

The work develops a framework to obtain large numbers of $A_\nu$-type singularities on algebraic surfaces by merging the ${\mathcal{J}}_d$ family with Belyi polynomials ${\mathcal{B}}^{(t)}_{d,\nu,\epsilon}(w)$. The singularity count is given by $N_0({\mathcal{J}})N_{-1}({\mathcal{B}},\nu) + N_{-1}({\mathcal{J}})N_1({\mathcal{B}},\nu)$, with explicit formulas for the constituent terms, leading to improved lower bounds $\mu_{A_\nu}(d)$ over prior results. The paper also provides explicit Jacobi-polynomial representations for several Belyi polynomials via $G_{a,b,c}(w)$, enabling closed-form, rational models and concrete examples of high singularity counts (e.g., degrees 9, 15, 21, and 36). These constructions yield large families of algebraic surfaces defined over $\mathbb{Q}$ with many singularities, contributing to extremal singularity problems and potential rigidity phenomena in projective geometry.

Abstract

We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit expressions in terms of classical Jacobi polynomials are obtained.

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

TL;DR

The work develops a framework to obtain large numbers of -type singularities on algebraic surfaces by merging the family with Belyi polynomials . The singularity count is given by , with explicit formulas for the constituent terms, leading to improved lower bounds over prior results. The paper also provides explicit Jacobi-polynomial representations for several Belyi polynomials via , enabling closed-form, rational models and concrete examples of high singularity counts (e.g., degrees 9, 15, 21, and 36). These constructions yield large families of algebraic surfaces defined over with many singularities, contributing to extremal singularity problems and potential rigidity phenomena in projective geometry.

Abstract

We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit expressions in terms of classical Jacobi polynomials are obtained.
Paper Structure (4 sections, 5 theorems, 23 equations, 6 figures)

This paper contains 4 sections, 5 theorems, 23 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Belyi polynomials
  3. Surfaces with many $A$ singularities
  4. Some explicit expressions in terms of classical Jacobi polynomials

Key Result

Lemma 2.1

There exist polynomials ${\mathcal{B}}^{(1)}_{d,\nu}(w)$, also denoted by ${\mathcal{B}}^{(1)}[n,m]$, with $k-1$ critical points of multiplicity $\nu$ with critical value $\zeta=-1$ and one critical point of multiplicity $\nu$ with critical value $\zeta=1$, where $k=3m+1, m\in {\Bbb{Z}}^{+}$ and

Figures (6)

  • Figure 1: Chebyshev polynomial $T_{9}(w)$ (top) and its associated plane tree (bottom).
  • Figure 2: Plane trees for ${\mathcal{B}}^{(1)}_{d,\nu}(w)$: (a) $d=12n+9, \nu=3n+2$, $n=0$ (left), $n=1$ (right), (b) $d=21n+36, \nu=3n+5$, $n=0$ (left), $n>0$ (right)
  • Figure 3: Plane trees for ${\mathcal{B}}^{(2)}_{d,\nu, \epsilon}(w)$: (a) $d=12n+15, \nu=3n+3, \epsilon=2$, $n=0$ (left) and $n>0$ (right), (b) $d=15n+21, \nu=3n+4, \epsilon=0$, $n=0$ (left) and $n>0$ (right)
  • Figure 4: Plane trees for (left) Belyi polynomials in Lemma 2.3 with $n=0$; (right) Belyi polynomials ${\mathcal{B}}_{a +c(b-1),c-1, a-1}(w)$ with explicit equations given by $G_{a,b,c}(w)$. The number of edges with label $y$ or $c-1$ is indicated in brackets above the suspension points.
  • Figure 5: Plane trees for ${\mathcal{B}}^{(3)}_{d,\nu, \epsilon}(w)$: (a) $d=12n+18, \nu=3n+4, \epsilon=1$, $n=0$ (left) and $n>0$ (right), (b) $d=18n+51, \nu=3n+8, \epsilon=0$, $n=0$ (left) and $n>0$ (right)
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • Proposition 3.2
  • proof