 fiber at $\bfq=0$ is the $\bbZ$-semigroup scheme $$V_{\mathbf{GL_0},0} := \bfq^{-0}(0)= \operatorname{Sing}_{0\times 0}\times\bbG_m,$$ where $\operatorname{Sing}_{0\times 0}$ represents the singular $0\times 0$-matrices. Let $\operatorname{Diag}_{0\times 0}\subset \operatorname{Mat}_{0\times 0}$ be the submonoid scheme of diagonal $0\times 0$-matrices, and set $$V_{\mathbf{\whT}}:=\operatorname{Diag}_{0\times 0}\times\bbG_m\subset V_{\mathbf{GL_0}}= \operatorname{Mat}_{0\times 0}\times\bbG_m.$$ This is a diagonalizable $\bbZ$-monoid scheme