On the existence of curves in K-trivial threefolds

TL;DR

This paper establishes a criterion for families of curves on nodal K-trivial threefolds to produce rigid curves upon smoothing, and demonstrates their existence on general Calabi-Yau threefolds with arbitrary degree and bounded genus.

Contribution

It introduces a new criterion for the emergence of rigid curves in smoothed K-trivial threefolds and proves their existence on general complete intersection Calabi-Yau threefolds.

Findings

Rigid curves exist on general Calabi-Yau threefolds of arbitrary degree.

The criterion links families of curves on nodal threefolds to their smoothing behavior.

Explicit bounds on genus for these rigid curves are provided.

Abstract

We give a criterion for a continuous family of curves on a nodal $K$-trivial threefold $X_0$ to contribute geometrically rigid curves to a general smoothing of $X_0$. As an application, we prove the existence of geometrically rigid curves of arbitrary degree and explicitly bounded genus on general complete intersection Calabi-Yau threefolds.