Algebraic $n$-Valued Monoids on $\mathbb{C}{P}^1$, Discriminants and Projective Duality

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.

revised definitions and updated results accordingly; some calculation errors and typos corrected