Algebraic n-Valued Monoids on ℂP¹, Discriminants and Projective Duality

2025/10/15 by Buchstaber, Victor, Kornev, Mikhail
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2510.14010

Abstract

In this work, we establish connections between the theory of algebraic n-valued monoids and groups and the theories of discriminants and projective duality. We show that the composition of projective duality followed by the Möbius transformation z↦ 1/z defines a shift operation \mathbbM_n(ℂP¹)↦ \mathbbM_n-1(ℂP¹) in the family of algebraic n-valued coset monoids \\mathbbM_n(ℂP¹)\_n∈ℕ. We also show that projective duality sends each Fermat curve xⁿ+yⁿ=zⁿ (n≥ 2) to the curve p_n-1(zⁿ; xⁿ, yⁿ)=0, where the polynomial p_n(z;x,y) defines the addition law in the monoid \mathbbM_n(ℂP¹). We solve the problem of describing coset n-valued addition laws constructed from cubic curves. As a corollary, we obtain that all such addition laws are given by polynomials, whereas the addition laws of formal groups on general cubic curves are given by series.

Citations

• On the Classification of n-Valued Monoids and Groups of Order 3
• n-Valued Groups, Kronecker Sums, and Wendt's Matrices
• Continuous two-valued discrete-time dynamical systems and actions of two-valued groups
• Algebraic 2-valued group structures on \mathbb P¹, Kontsevich-type polynomials, and multiplication formulas, I
• Product formulas for the Higher Bessel functions
• Generalization and Geometrization of the Kowalevski top

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