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 $X={\mathbb{P}}^m\times {\mathbb{P}}^n$ and for this, we define a partial order on the bidegrees of the bihomogeneous forms. As a consequence of this construction, we computer explicitly the Hilbert scheme for curves of genus 7 and 8 listed in \cite{MUK95} and \cite{MUKIDE03} that are complete intersections. Finally, we construct the coarse moduli space of complete intersections in ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

19 pages