3 problems

Exceptional-set characterization. The exceptional set $\mathcal{E}(A)$ is empty if and only if the toric ideal $I_A$ is Cohen–Macaulay.

Sturmfels's conjecture. The exceptional set is empty if and only if $I_A$ is Cohen–Macaulay. For Cohen–Macaulay $I_A$, the rank equals $\operatorname{vol}(A)$ for every parameter,…

Gevrey-slope conjecture. The map $\beta\mapsto {\operatorname{d}_s}(A,\beta)$ is upper-semicontinuous. Moreover,