Algebraic magnetism invariants of a double scalar action on the projective plane

This document is an expanded version of the notes from a talk at the $\mathit{\text{Arithmetic}}\mathit{\text{and}}\mathit{\text{Algebraic}}\mathit{\text{Geometry}}\mathit{\text{Week}}$ 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}.