On Hilbert scheme of complete intersection on the biprojective

TL;DR

This paper constructs the Hilbert scheme of complete intersections in biprojective space, explicitly computes it for certain genus 7 and 8 curves, and develops the moduli space for complete intersections in \\mathbb{P}^1\\times\\mathbb{P}^1.

Contribution

It introduces a method to construct the Hilbert scheme of complete intersections in biprojective space and explicitly computes it for specific genus curves.

Findings

Explicit Hilbert scheme for genus 7 and 8 complete intersection curves

Construction of the coarse moduli space in P^1 x P^1

Definition of a partial order on bidegrees of bihomogeneous forms

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 $\mathbb{P}^1\times\mathbb{P}^1$.