The relevant range is the values of \((p,q)\) for which the Betti numbers \(\beta_{p,q}(2,b;d)\) are non-zero and not determined solely by the multigraded Hilbert series of \(S(2,b;d)\). This is the range in which we must compute the multigraded Betti numbers by computing the ranks of the accompanying differentials.

For \(\mathbb{P}^{2}\) the relevant range can alternatively be characterized as the set of \((p,q)\) where \(K_{p,q}(2,b;d)\neq0\) and \(K_{p-1,q+1}(2,b;d)\neq0\) or \(K_{p+1,q-1}(2,b;d)\neq0\). A mixture of various results (see Rmk) gives shows \([K_{p,q}(2,b;d)\neq0\) for all \(p\) in the following range: \[ \binom{m+d}{d}-\binom{m+d-r-1}{m}-m\leq p \leq \binom{2+d}{2}+\binom{2-m+r}{2-m}-\binom{2-m+d}{2-m}-m-1 \] where \(m\) and \(r\) are the quotient and remainder of \(qd+b\) divided by \(d-1\). As these bounds are known to be sharp they allow us to compute the relevant range easily. For example, if \(b=0\) and \(d=0\) the relevant range is \(\{(14,1), (15,1), (13,2), (14,2)\}\).