Curves in Segre threefolds

TL;DR

This paper investigates the structure and classification of low-degree Cohen-Macaulay curves in Segre threefolds, analyzing their Hilbert schemes and related moduli spaces to understand their geometric properties.

Contribution

It provides new results on the irreducibility and connectedness of Hilbert schemes and moduli spaces of curves and sheaves in Segre threefolds with different Picard numbers.

Findings

Identified irreducible components of Hilbert schemes for low-degree curves.

Proved irreducibility of certain moduli spaces of stable sheaves.

Applied methods to Segre threefolds with Picard number two.

Abstract

We study locally Cohen-Macaulay curves of low degree in the Segre threefold with Picard number three and investigate the irreducible and connected components respectively of the Hilbert scheme of them. We also discuss the irreducibility of some moduli spaces of purely one-dimensional stable sheaves and apply the similar argument to the Segre threefold with Picard number two.