Algebraic $n$-Valued Monoids on $\mathbb{C}P^1$, Discriminants and Projective Duality
Victor Buchstaber, Mikhail Kornev
TL;DR
The paper builds an algebro-geometric framework linking algebraic $n$-valued monoids/groups on $\mathbb{C}P^1$ with discriminants and projective duality. It shows that duality composed with the Möbius map induces a shift $\mathbb{M}_n(\mathbb{C}P^1)\to\mathbb{M}_{n-1}(\mathbb{C}P^1)$ and connects curve duality to the polynomials $p_n$ that encode $n$-valued addition laws, including Fermat curves via $p_{n-1}(z^n;x^n,y^n)=0$. The work classifies coset addition laws arising from cubic models, yielding explicit polynomial laws for $2$-, $3$-, $4$-, and $6$-valued structures, and analyzes nodal and cuspidal degenerations, with doubling/iterating phenomena governed by discriminants. These results provide a purely algebro-geometric perspective on $n$-valued monoids/groups and their interplays with discriminants, duality, and elliptic geometry, with explicit constructions and isomorphism classes determined by $j$-invariants and singularity type.
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 $z\mapsto 1/z$ defines a shift operation $\mathbb{M}_n(\mathbb{C}P^1)\mapsto \mathbb{M}_{n-1}(\mathbb{C}P^1)$ in the family of algebraic $n$-valued coset monoids $\{\mathbb{M}_{n}(\mathbb{C}P^1)\}_{n\in\mathbb{N}}$. We also show that projective duality sends each Fermat curve $x^n+y^n=z^n$ $(n\ge 2)$ to the curve $p_{n-1}(z^n; x^n, y^n)=0$, where the polynomial $p_n(z;x,y)$ defines the addition law in the monoid $\mathbb{M}_n(\mathbb{C}P^1)$. 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.