  We investigate the space of solutions to certain $A$-hypergeometric
$\mathscr{D}$-modules, which were defined and studied by Gelfand, Kapranov, and
Zelevinsky. We show that the solution space can be identified with certain
relative cohomology group of the toric variety determined by $A$, which
generalizes the results of Huang, Lian, Yau, and Zhu. As a corollary, we also
prove the existence of rank one points for Calabi--Yau complete intersections
in toric varieties.
