On Hilbert scheme of complete intersection on the biprojective

Abstract The goal of this paper is to construct the Hilbert scheme of complete intersections in the biprojective space $$ X = \mathbb {P}^m \times \mathbb {P}^n $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> . 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 $$ \mathbb {P}^m \times \mathbb {P}^n $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> .