The circuit ideal of a vector configuration

Given a configuration A={a1,…,an}⊂Zd, a basis ideal of A is an ideal where B spans the lattice . Our main interest is to understand when the toric ideal, IA, of A equals a basis ideal JB with radical IA. The circuit ideal, JCA, of A is an example of such a basis ideal.We study such a JB in relation to IA from various algebraic and combinatorial perspectives with a special focus on JCA. We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of JB and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that the embedded primes of JB are indexed by certain faces of the cone spanned by A. We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to IA and JCA. The Gröbner fan of JCA is shown to refine that of IA when the codimension of the ideals is at most two.