Macaulay bound admissibility for positive-dimensional fibers

Determine whether, for a bihomogeneous ideal I ⊂ k[x_0,…,x_n,y_0,…,y_m] generated by polynomials f_i of bidegrees (a_i,b_i) and with finite projection π(V(I)) ⊂ P^n but possibly positive-dimensional fibers over π, every bidegree (a,b) ≥ ∑_i(a_i,b_i) − (n,m) is an admissible bidegree in the sense of Definition 2.1 (i.e., the Hilbert function stabilizes in the x-direction at (a,b) and there exists a linear form h of bidegree (1,0) with HF_{R/(I,h)}(a,b)=0).

Background

The paper introduces admissible bidegrees (Definition 2.1) for bihomogeneous ideals I ⊂ k[x,y] as the degrees where the Hilbert function stabilizes in the x-direction and a suitable linear form h annihilates the corresponding graded component, enabling the construction of multiplication maps to recover the zero-dimensional projection π(V(I)).

They prove that when V(I) defines finitely many points (i.e., V(I) is zero-dimensional), the multihomogeneous Macaulay bound—(a,b) ≥ ∑_i(a_i,b_i) − (n,m) for generators f_i of bidegrees (a_i,b_i)—guarantees admissibility (Theorem 4.1). For the more general setting where π(V(I)) is finite but V(I) may have positive-dimensional fibers, the authors could not establish whether the same Macaulay bound remains sufficient, and they instead provide a looser bound via generalized Koszul complexes (Theorem 4.2).