Algebraic magnetism invariants of a double scalar action on the projective plane
Arnaud Mayeux
TL;DR
The paper analyzes a rank-two diagonalizable group action on the projective plane via Algebraic Magnetism, explicitly computing magnets, pure magnets, and attractors for the double scalar action on P^2. It establishes a bijection between pure magnets and additively stable subsets of the action’s weight set, yielding a complete 29-entry stratification and a magnetic analogue of the Bialynicki-Birula decomposition. It further introduces lambdafiable magnets, linking many attractors to cocharacters and clarifying which strata arise from G_m-actions. Finally, it reports on new theoretical advances from BM, proving finiteness of pure magnets and advancing atlas-type results for diagonalizable group actions on finite-presentation schemes.
Abstract
This document is an expanded version of the notes from a talk at the \textit{Arithmetic and Algebraic Geometry Week} conference, which took place in Iasi in September 2025. In this note, we compute the pure magnets (certain semigroups) and the associated attractors for a double scalar action of $\mathbb{G}_m^2$ on $\mathbb{P}^2$. This is mostly expository and provides a non-affine example illustrating the invariants of Algebraic Magnetism in a simple and visual case. Nevertheless, we introduce the notion of lambdafiable magnets, in the general setting, to relate certain magnets to cocharacters. Finally we announce recent advanced results on Algebraic Magnetism obtained in \cite{BM} and solving positively some conjectures stated in \cite{Ma}.