Algebraic Magnetism

Abstract We introduce Algebraic Magnetism, a theory of attractors associated to magnets within the framework of actions of diagonalizable monoid schemes on algebraic spaces. For a diagonalizable monoid scheme $$A(M)_S$$ A ( M ) S acting on an algebraic space X , we introduce for any submonoid N of M an attractor space $$X^N$$ X N . We then investigate and study various aspects of attractors associated to monoids. This leads to the notion of pure magnets, which encode the combinatorial structure underlying attractor phenomena in terms of semigroups. In the affine case, we give an explicit description of attractors in terms of graded algebra, yielding representability as closed subspaces in this case. Using descent methods, technical preliminary results, and fixed-point-reflecting atlases, we prove representability for general algebraic spaces. We establish compatibility of attractors with fiber products, base change, faces of monoids, intersections of submonoids, subgroups, dilatations, and many other operations. We study pure magnets and prove, and conjecture, finiteness results under suitable hypotheses. In this way, Algebraic Magnetism provides invariants to study diagonalizable actions in algebraic geometry.