The goal of this paper is to construct the Hilbert scheme of complete intersections in the biprojective space \( X = \mathbb {P}^m \times \mathbb {P}^n \) . For this purpose, we define a partial order on the bidegrees of the bihomogeneous forms. As a consequence of this construction, we provide an explicit computation of the Hilbert scheme for curves of genus 7 and 8 listed in [1] and [11] that are complete intersections. Finally, we construct the coarse moduli space of canonical complete intersections in \( \mathbb {P}^m \times \mathbb {P}^n \) .
