On smooth and isolated curves in general complete intersection Calabi-Yau threefolds

TL;DR

This paper develops new methods to determine when curves in certain Calabi-Yau threefolds can deform into smooth isolated curves, leading to the discovery of new such curves.

Contribution

It introduces novel techniques to verify node conditions on linear systems, enabling the construction of new smooth isolated curves in Calabi-Yau threefolds.

Findings

New criteria for node conditions on linear systems

Construction of previously unknown smooth isolated curves

Enhanced understanding of curve deformation in Calabi-Yau threefolds

Abstract

Recently Knutsen found criteria for the curves in a complete linear system $|\mathcal{L}|$ on a smooth surface $X$ in a nodal K-trivial threefold $Y_0$ to deform to a scheme of finitely many smooth isolated curves in a general deformation $Y_t$ of $Y_0$. In this article we develop new methods to check whether the set of nodes of $Y_0$ imposes independent conditions on $|\mathcal{L}|$. As an application, we find new smooth isolated curves in complete intersection Calabi-Yau threefolds.