Core Coordinates and Negative Support in A-Hypergeometric Systems

For a homogeneous $A$-hypergeometric system, let $K_v$ be the intersection of the negative support family for a fake exponent $v$, and let $C_w$ be the set of indices of variables that do not occur in any minimal monomial generator of ${\mathrm{in}}_w(I_A)$. We prove that $K_v=\mathrm{nsupp}(v)\cap C_w$ holds for every parameter and every fake exponent if and only if $\mathrm{Cone}(A_{C_w})$ meets the interior of $\mathrm{Cone}(A)$. We give equivalent criteria using the reduced Gröbner basis and the associated regular triangulation. When the condition fails, we construct an infinite family of fake exponents for which the equality fails.