We investigate the solution space to certain A -hypergeometric 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 complete intersections in toric varieties. [ABSTRACT FROM AUTHOR]