SyzygyData

Syzygies of $$S(1,b;d)$$: Although these results are well known, we include data for $$\mathbb P^1$$ for convenience. Click on the pair $$(d,b)$$ below to see the associated Betti tables or scroll to the bottom of the page to see information regarding the relevant range of computation and ways we have verified our work.

 $$(d,b)$$ 2 3 4 5 6 7 8 9 10 0 (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0) (9,0) (10,0) 1 (2,1) (3,1) (4,1) (5,1) (6,1) (7,1) (8,1) (9,1) (10,1) 2 — (3,2) (4,2) (5,2) (6,2) (7,2) (8,2) (9,2) (10,2) 3 — — (4,3) (5,3) (6,3) (7,3) (8,3) (9,3) (10,3) 4 — — — (5,4) (6,4) (7,4) (8,4) (9,4) (10,4) 5 — — — — (6,5) (7,5) (8,5) (9,5) (10,5) 6 — — — — — (7,6) (8,6) (9,6) (10,6) 7 — — — — — — (8,7) (9,7) (10,7) 8 — — — — — — — (9,8) (10,8) 9 — — — — — — — — (10,9)

$$\text{pd} S(1,b;d)=\binom{d+1}{1}-1$$

The relevant range is the values of $$(p,q)$$ for which the Betti numbers $$\beta_{p,q}(1,b;d)$$ are non-zero and not determined solely by the multigraded Hilbert series of $$S(1,b;d)$$. This is the range in which we must compute the multigraded Betti numbers by computing the ranks of the accompanying differentials.

For $$\mathbb{P}^{1}$$ the relevant range is trivial as there is never any overlap between the rows of the Betti table, and so the Betti table is entirely determined from the Hilbert series.

Verifications

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

• For all $$d\geq0$$ and all $$b< d$$ the minimal resolution of $$S(1,b;d)$$ is known to be given by an Eagon-Northcott complex. (For example, see Corollary 6.2 in  for a precise statement when $$b=0$$.) From this it is possible to recover the graded, multigraded, and Schur graded Betti numbers. Our computations agree with these.