Algebraic magnetism stratification of the double scalar action on the projective plane

Abstract: We determine the pure magnets (certain monoids) for the double scalar action of $\mathbb{G}_m^2$ on $\mathbb{P}^2$, and compute the associated attractors, yielding a canonical, fully explicit stratification. This provides a concrete non-affine example illustrating the invariants of Algebraic Magnetism.