Apolarity for border cactus decompositions

Abstract: The border apolarity technique was introduced in our earlier work for secant varieties over complex numbers. We extend the theory to cactus varieties of toric varieties over any algebraically closed field. A border cactus decomposition is a mulithomogeneous ideal in the Cox ring (also called the total coordinate ring) of the toric variety that witnesses that a given point is in a specific cactus variety. The definition of such witness uses apolarity and we describe the set of ideals that are credible witnesses for this purpose in terms of a correspondence between the usual Hilbert scheme (parametrising all closed subschemes of the toric variety) and the multigraded Hilbert scheme (parametrising all multihomogeneous ideals in the Cox ring). We also take this opportunity to extend the border apolarity to linear subspaces (in non-border setting, this is equivalent to simultaneous decompositions).