On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds

TL;DR

This paper establishes criteria for deforming families of curves in nodal Calabi-Yau threefolds to isolated smooth curves in general deformations, demonstrating their existence with bounded genus and unbounded degree.

Contribution

It provides new deformation criteria for curves in Calabi-Yau threefolds and proves the existence of isolated smooth curves with specific properties.

Findings

Criteria for deforming families of curves to isolated smooth curves

Existence of smooth isolated curves of bounded genus and unbounded degree in Calabi-Yau threefolds

Extension of previous results by Clemens and Kley

Abstract

Building on results of Clemens and Kley, we find criteria for a continuous family of curves 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 $Y_0$. As an application, we show the existence of smooth isolated curves of bounded genera and unbounded degrees in Calabi-Yau complete intersections threefolds.