SyzygyData

Syzygies of $$S(2,b;d)$$: 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. Green signifies we have a completed our computations, and have all the graded, multigraded, and Schur graded Betti numbers. Yellow indicates we have not yet finished (or in a few of the large cases started) our computations yet. There may be errors for these entries that will be corrected with further computation. Red signifies we are aware of errors in these examples, and working to correct them. Stay tuned!

 $$(d,b)$$ 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)

$$\text{pd} S(2,b;d)=\binom{d+2}{2}-2$$

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

Verifications

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.