\((d,b)\) | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

0 | (2,0) | (3,0) | (4,0) | (5,0) | (6,0) | (7,0) | (8,0) |

1 | (2,1) | (3,1) | (4,1) | (5,1) | (6,1) | (7,1) | (8,1) |

2 | — | (3,2) | (4,2) | (5,2) | (6,2) | (7,2) | (8,2) |

3 | — | — | (4,3) | (5,3) | (6,3) | (7,3) | (8,3) |

4 | — | — | — | (5,4) | (6,4) | (7,4) | (8,4) |

5 | — | — | — | — | (6,5) | (7,5) | (8,5) |

6 | — | — | — | — | — | (7,6) | (8,6) |

7 | — | — | — | — | — | — | (8,7) |

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 \(\{(14,1), (15,1), (13,2), (14,2)\}\).

Here we record ways in which our computations have been checked/verified:

- For \(d\leq 4\) the graded, multigraded, and Schur graded Betti numbers of \(S(2,b;d)\) can be computed exactly via symbolic Gröbner methods. Our computations agree with these.
- For \(d=4\) and \(d=4\) the graded Betti numbers of \(S(2,0;d)\) were computed using a mixture of theoretical results and symbolic computations in [7]. Our computations agree with these.
- For \(d\leq 6\) the graded and multigraded Betti numbers of \(S(2,0;d)\) were computed via a similar numerical approach (working over a finite field) in [1]. Our computations agree with these.

Non-vanishing in these ranges is the contents of Theorem 6.1 of [3] and Theorem 2.1 of [2]. (Both of these theorems are vastly more general providing similar statements for all \(n\).) The sharpness of these bounds follows from Remark 6.5 in [3] for \(q=0,2\) and for \(q=1\) from Theorem 2.2 in [9] and Theorem 2.c.6 in [8].