Polarized K3 surfaces with an automorphism of order 3 and low Picard number

2023/06/30 by Dino Festi, Festi, Dino
Arts and Humanities · Computer Science · Mathematics · #14J10 #14J28 #14J50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Historical Studies and Socio-cultural Analysis

paper · pdf · doi:10.48550/arxiv.2306.17539

openalex publication_date 2023/06/30 · openalex created_date 2023/07/04 · openalex updated_date 2026/07/28

Abstract

In this paper, for each d>0, we study the minimum integer h_3,2d∈ ℕ for which there exists a complex polarized K3 surface (X,H) of degree H²=2d and Picard number ρ(X):=\textrmrank \textrmPic X = h_3,2d admitting an automorphism of order 3. We show that h_3,2∈\ 4,6\ and h_3,2d=2 for d>1. Analogously, we study the minimum integer h^*_3,2d∈ ℕ for which there exists a complex polarized K3 surface (X,H) as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of X. We show that h^*_3,2d is equal to 2 if d>1 and equal to 6 if d=1. We provide explicit examples of K3 surfaces defined over ℚ realizing these bounds.

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