In the proof of theorem from c to a show that the subcases

Question

Question: In the proof of Theorem (from (c) to (a)), show that the subcases (A₃), (B₁), and (B₃) cannot occur.

Theorem: If K[F] is homogeneous and normal and PF is its toric ideal, then the following are equivalent:

(a) PF is generated by a finite set of circuits.

(b) PF is generated by a finite set of circuits with a square-free term.

(c) Every unbalanced circuit of PF has a connector which is a linear combination (with coefficients in K[t]) of circuits of PF with a square-free term.