[Paper Review] Picard numbers of complex Delsarte surfaces with only isolated ADE-singularities

Bas Heijne|arXiv (Cornell University)|Dec 20, 2012

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper classifies all complex Delsarte surfaces of degree $ n \geq 6 $ with only isolated ADE singularities, proving there are at most 83 such surfaces up to isomorphism. It provides a closed-form formula for the Picard number of each family, derived using the Lefschetz number via the algorithm of Delsarte and Shioda, with explicit computation based on the exponent matrix and its inverse, and confirms three maximal surfaces at degree $ n \geq 5 $.

ABSTRACT

We give a classification of all Delsarte surfaces with only ADE singularities. Using this we give closed formulas for the Picard numbers of such surfaces.

Motivation & Objective

• To classify all complex Delsarte surfaces of degree $ n \geq 6 $ with only isolated ADE singularities up to isomorphism.
• To compute the Picard number for each such surface using the Lefschetz number formula from Delsarte's algorithm.
• To identify and characterize maximal Delsarte surfaces where the Picard number achieves the Hodge theoretic upper bound.
• To provide explicit closed-form formulas for the Picard number depending only on the degree $ n $, valid for all such families.
• To extend prior results on non-singular Delsarte surfaces to the singular ADE case, particularly for higher degrees.

Proposed method

• The exponent matrix $ A $ of the defining polynomial is constructed from the monomials of the surface.
• The inverse matrix $ A^{-1} $ is used to compute three vectors $ \tilde{v}, \tilde{w}, \tilde{u} $ via multiplication with row vectors $ e_1, e_2, e_3 $.
• These vectors generate a finite $ \mathbb{Z} $-module $ L \subseteq (\mathbb{Q}/\mathbb{Z})^4 $, with $ L_0 $ and $ L_1 $ defined as subsets where at least one or all coordinates are zero, respectively.
• The set $ \Lambda \subseteq L_1 $ is defined by conditions on the order and fractional parts of scalar multiples of elements in $ L_1 $, and its cardinality gives the Lefschetz number $ \lambda = \#\Lambda $.
• The Picard number is computed as $ \rho = b_2 - \lambda $, where $ b_2 = n^3 - 4n^2 + 6n - 2 $ for surfaces of degree $ n $ with ADE singularities.
• The classification relies on analyzing the structure of $ L_1 \setminus \Lambda $, which splits into decomposable, regularly indecomposable, and exceptional (irregular indecomposable) elements, with the latter being finite and explicitly known.

Experimental results

Research questions

• RQ1 How many Delsarte surfaces of degree $ n \geq 6 $ with only isolated ADE singularities exist up to isomorphism?
• RQ2 What is the closed-form formula for the Picard number of each such surface, depending only on the degree $ n $?
• RQ3 Which Delsarte surfaces are maximal, i.e., achieve equality in the Hodge number bound $ \rho \leq h^{1,1} $?
• RQ4 How do the Lefschetz number and Picard number behave across different families of Delsarte surfaces with ADE singularities?
• RQ5 What is the role of the exponent matrix and its inverse in computing the Picard number via the $ \Lambda $-set construction?

Key findings

• There are at most 83 Delsarte surfaces of degree $ n \geq 6 $ with only isolated ADE singularities up to isomorphism.
• For each of these 83 families, a closed-form formula for the Picard number is provided, depending only on the degree $ n $, as detailed in Appendix A.
• The three maximal Delsarte surfaces of degree $ n \geq 5 $ are: $ X^3YZ + Y^3ZU + XZ^3U + XYU^3 = 0 $, $ X^5Y + XY^5 + Z^5U + ZU^5 = 0 $, and $ X^6 + Y^6 + Z^6 + U^6 = 0 $.
• The Picard number is computed as $ \rho = b_2 - \lambda $, where $ b_2 = n^3 - 4n^2 + 6n - 2 $, and $ \lambda = \#\Lambda $, with $ \Lambda $ derived from the exponent matrix and its inverse.
• The set $ L_1 \setminus \Lambda $ is partitioned into decomposable, regularly indecomposable, and exceptional (irregular indecomposable) elements, with the latter being finite and explicitly known with cardinality 22080.
• The formula for the Picard number in each case includes correction terms via Kronecker delta functions $ \delta_{a,b} $, which account for special cases depending on the degree $ n $.