Rational curves in CICY's in products of two projective spaces

TL;DR

This paper proves the finiteness of certain smooth rational curves on Calabi-Yau threefolds formed as complete intersections in products of two projective spaces, extending Clemens' conjecture to low bidegrees.

Contribution

It establishes the finiteness of smooth rational curves with low bidegree and nondegenerate projections on specific CICY threefolds in products of projective spaces.

Findings

Number of such rational curves is finite in low bidegrees.

Finiteness holds even when degenerate rational curves form positive-dimensional families.

Extends Clemens' conjecture to a new class of Calabi-Yau threefolds.

Abstract

Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More precisely, we prove that the number of smooth rational curves on $Y$ with low bidegree and with nondegenerate birational projection is at most finite (even in cases in which positive dimensional families of degenerate rational curves are known).