Implicit equations for general multi-Rees algebras

Determine explicit generators, or equivalently implicit equations, for the presentation ideals of multi-Rees algebras and their special fiber rings associated with arbitrary collections of homogeneous ideals.

Background

For a collection of homogeneous ideals I_1, ..., I_r in a polynomial ring, the multi-Rees algebra and its special fiber ring can be represented as quotients by presentation ideals. When the ideals are generated by monomials of the same degree, these presentation ideals are generated by multigraded binomials and define toric varieties when they are prime.

The paper notes that finding an explicit description of these presentation ideals is important both for understanding the associated blowup algebras and for applications such as equilibrium solutions of chemical reaction networks. The authors solve this problem for collections of ideals arising from standardizable Ferrers diagrams, but state that the general implicit-equation problem remains unresolved.