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

Abstract: Algebraic surfaces in the complex projective space with a high number of A-type singularities have been presented in a paper. We extend the construction in order to obtain lower bounds for the maximal number of A singularities for certain additional cases.

• The paper introduces a recursive combinatorial method to explicitly construct algebraic surfaces with record lower bounds for A-singularities.
• It systematically employs Belyi polynomials and planar tree transformations to control critical point structures and singularity multiplicities.
• The approach produces infinite families of surfaces with asymptotically improved singularity counts compared to previous constructions.

Lower Bounds for the Maximal Number of A-singularities in Algebraic Surfaces

Introduction and Context

The enumeration of singularities of $A$-type (also called simple or ordinary $A_\nu$-singularities) on complex projective algebraic surfaces of fixed degree $d$ is a central question in classical and modern algebraic geometry. While upper bounds for the maximal number of such singularities are known in certain cases, providing sharp or improved lower bounds relies on explicit constructions of surfaces exhibiting many singularities of the prescribed type. This paper extends previous frameworks for such constructions, particularly the techniques in Escudero [esc26], focusing on degrees $d=3q$, and introduces systematic methods to generate new infinite families of surfaces with provably high numbers of $A_\nu$ singularities by manipulating associated tree combinatorics of polynomials.

Families of Polynomials with Controlled Critical Structure

The crux of the construction is the interplay between univariate polynomials with two critical values (Belyi polynomials), bicolored planar trees encoding the distribution of critical points, and explicit combinatorial generation rules. The language of trees, with alphabets given by transformation rules ($\alpha$, $\beta$, etc.), allows one to recursively build polynomials with a prescribed mix of critical points—controlling both their multiplicity and their critical values. Given an initial tree encoding a Belyi polynomial as in [esc26], sequences of transformations yield more elaborate combinatorial types, each corresponding to a new polynomial $\mathcal{G}$ with controlled critical behavior.

A key technical device is the introduction of a formal language $\mathcal{L}_E$, whose admissible words correspond to sequences of tree transformations producing polynomials satisfying condition $(E)$: for a given $A_\nu$0 (the desired singularity type), tree-derived polynomials $A_\nu$1 have precisely $A_\nu$2 critical points of multiplicity $A_\nu$3 with value $A_\nu$4 and the exact complementary count (up to floor operations) for critical value $A_\nu$5.

Two principal families of surface-defining polynomials emerge: $A_\nu$6, constructed via alphabets $A_\nu$7 (with $A_\nu$8 encoding specific subclasses), and, for more intricate cases, a larger alphabet including $A_\nu$9, $d$0, $d$1, $d$2, and $d$3. These recursive rules yield infinite sequences for certain $d$4, especially $d$5, efficiently generating polynomials of arbitrarily large degree with maximal numbers of high-multiplicity critical points.

Explicit Construction of Surfaces with Many $d$6 Singularities

Surfaces are constructed in affine form as zero loci of the sum $d$7, where $d$8 are well-structured bivariate polynomials with a rich, explicit critical point structure and $d$9 are rescaled versions of the $d=3q$0 polynomials. The key to the multiplicity count is identifying loci in $d=3q$1 where critical points of both summands align to enhance the singularity.

The method accounts for nodes ($d=3q$2-singularities), cusps ($d=3q$3), and higher $d=3q$4 types, proving the existence of surfaces with the following lower bound for the number of $d=3q$5 singularities:

$d=3q$6

and, in the case $d=3q$7,

$d=3q$8

for $d=3q$9.

These constructions consistently yield singularity counts exceeding prior constructions in select ranges, and always meet or surpass bounds in [lab06]. The jump in cusp count for $A_\nu$0 surfaces, with the language $A_\nu$1 being infinite, demonstrates the systematic extensibility of the approach.

Comparison to Previous Methods and Improvements

The surfaces previously constructed using folding polynomials associated with root lattices (e.g., $A_\nu$2) and special Belyi polynomials [lab06] achieved substantial numbers of $A_\nu$3 singularities. However, the framework of this paper introduces an additional degree of freedom by permitting a more general and recursive tree transformation scheme, allowing the generation of polynomials with an increased number of high multiplicity critical points.

For all $A_\nu$4, the derived surfaces have at least as many $A_\nu$5 singularities as the best previously known constructions, and for various values of $A_\nu$6 and $A_\nu$7, they explicitly yield higher counts—for example, at least $A_\nu$8 more cusps for certain degrees $A_\nu$9 compared to [lab06].

Theoretical Implications and Prospects

This methodology exposes a new systematic route to approach the extremal function $\alpha$0, the maximal number of $\alpha$1 singularities on a degree $\alpha$2 surface. While upper bounds remain challenging, explicit recursive lower bounds with combinatorial control suggest new directions for both computational and theoretical advances. The tree-based approach, coupled with formal languages for admissible transformations, may enable future progress on related problems: surfaces with prescribed singularity spectra, extensions to higher dimensions (hypersurfaces, threefolds), and connections with dessins d’enfants, Galois actions, and the arithmetic of polynomials with two critical values.

Open questions include sharpness of bounds for large $\alpha$3, identification of possible symmetry constraints, and the explicit description of singular loci in geometric terms. The apparent infinitude of the admissible transformation language for certain $\alpha$4 suggests deeper algebraic and combinatorial structure still to be explored.

Conclusion

This work advances the construction of complex projective surfaces with many $\alpha$5-singularities by systematically encoding and manipulating polynomial critical data using tree combinatorics and formal languages. The approach not only provides new, improved lower bounds for the maximal number of $\alpha$6 singularities for infinite families of degrees, but also introduces a recursive and extensible technical toolset for the construction of singularity-rich algebraic surfaces, with implications for both explicit examples and asymptotic lower bounds (2604.15060).