Weighted surfaces with maximal Picard number

2025/06/16 by Louis Esser, Jennifer Li, Esser, Louis +1 · 1 citation
Engineering · Mathematics · #14C22 #14J27 #14J70 #Advanced Differential Equations and Dynamical Systems #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2506.14037

openalex publication_date 2025/06/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

An algorithm due to Shioda computes the Picard number for certain surfaces which are defined by a single equation with exactly four monomials, called Delsarte surfaces. We consider this method for surfaces in weighted projective 3-space with quotient singularities. We give a criterion for such a weighted Delsarte surface X to have maximal Picard number. This condition is surprisingly related to the automorphism group of X. For every positive integer s, we find a weighted Delsarte surface with geometric genus s and maximal Picard number. We show that these examples are elliptic surfaces, proving that elliptic surfaces of maximal Picard number and arbitrary geometric genus may be embedded as quasismooth hypersurfaces in weighted projective space.

Citations

• Rational weighted projective hypersurfaces
• Rationality of weighted hypersurfaces of special degree
• Notions of maximality for integral lattice-free polyhedra: the case of\n dimension three
• Some surfaces with maximal Picard number
• The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces
• The Maximal Rank of Elliptic Delsarte Surfaces
• Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three
• 3-Dimensional Lattice Polytopes Without Interior Lattice Points

Cited by

• Finite abelian groups acting on rationally connected threefolds II: groups of K3 type

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