Reducible Calabi-Yau threefolds with countably many rational curves

TL;DR

This paper presents examples of reducible Calabi-Yau threefolds with infinitely many rational curves, showing that such curves can exist without sweeping out a surface, challenging previous assumptions.

Contribution

It introduces a new class of reducible Calabi-Yau threefolds with countably many rational curves, expanding understanding of their geometric properties.

Findings

Existence of rational curves in reducible CY threefolds without sweeping out a surface

Infinitely many degrees of stable maps occur in these examples

Examples are of d-semistable threefolds with two irreducible components

Abstract

We give a class of examples of reducible (d-semistable) threefolds of CY type with two irreducible components for which (it is reasonably easy to prove that) no family of admissible genus zero stable maps sweeps out a surface, yet such stable maps occur in infinitely many degrees.