An example of a non-associative Moufang loop of point classes on a cubic surface

2021/04/11 by Dimitri Kanevsky, Kanevsky, Dimitri · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Associative property #Combinatorics #Commutative property #Computer science #Equivalence relation #Extension (predicate logic) #FOS: Mathematics #Geometry #Loop (graph theory) #Mathematics #Mathematics and Applications #Modulo #Number Theory (math.NT) #Pure mathematics #Quadratic equation #Surface (topology) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2104.05118

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2021/04/11 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let V be a cubic surface defined by the equation T₀³+T₁³+T₂³+θT₃³=0 over a quadratic extension of 3-adic numbers k=ℚ₃(θ), where θ³=1. We show that a relation on a set of geometric k-points on V modulo (1-θ)³ (in a ring of integers of k) defines an admissible relation and a commutative Moufang loop associated with classes of this admissible equivalence is non-associative. This answers a problem that was formulated by Yu. I. Manin more than 50 years ago about existence of a cubic surface with a non-associative Moufang loop of point classes.

Cited by

• R-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

Related

• Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral
• Relations in the tautological ring of the moduli space of K3 surfaces
• Noncommutative marked surfaces
• Persistence of the Brauer-Manin obstruction on cubic surfaces
• Cubic surfaces with a Galois invariant pair of Steiner trihedra