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

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).