The index-weighted vanishing conjecture for semi-ample growth

Let $S$ be the bihomogeneous polynomial ring and let $\overline{S}$ denote the quotient defined in the paper. Fix $\mathbf n=(n_1,n_2)\in \mathbb{Z}^2_{\geq 1}$, set $\mathbf d=\mathbf 1$, and write $\overline{S}_{\mathbf a,k}$ for the component of bi-degree $\mathbf a$ and index weighted degree $k$. Given $\mathbf a=(a_1,a_2)\in\mathbb{Z}^2_{\geq 0}$ and $k\in\mathbb{Z}_{\geq 0}$, the index-weighted vanishing conjecture. $\dim \overline{S}_{\mathbf a,k}=0$ if and only if at least one of the following holds: $a_1\geq 1$ and $a_2\geq n_1+1$; $a_2\geq 1$ and $a_1\geq n_2+1$; $0\leq k\leq a_1a_2-1$; or $k\geq a_1n_1+(n_2-a_1)a_2+1$. Moreover, if $k=a_1a_2$ or $k=a_1n_1+(n_2-a_1)a_2$, then $\dim \overline{S}_{\mathbf a,k}=1$. This conjecture is intended to characterize exactly the multidegrees and index weighted degrees for which the relevant graded component vanishes, complementing the paper’s partial answer to the ideal-membership problem for $\mathfrak{R}$.