There has been considerable interest in the study of curves, especially rational curves, on threefolds with trivial canonical bundle. Under the name of Calabi–Yau threefolds this class of algebraic varieties has also attracted the interest of mathematical physicists. One special case which has been studied extensively is that of complete intersection threefolds with trivial canonical bundle—for example, hypersurfaces of degree 5 in P4.
A paper of P. Griffiths [9] showed that a general hypersurface X of degree 5 in P4 contains a pair of lines L1 and L2 such that no multiple of the cycle ξ = [L1] - [L2] in CH2(X) is rationally equivalent to zero. This was taken further by C. H. Clemens [4] who found an infinite collection of rational curves on X and showed that these curves generate a subgroup of CH2(X) which has infinite rank.
In this work we constructed such a collection of rational curves for other complete intersection threefolds. The results of C. H. Clemens were generalised to these cases also.
Theorem 6.1. Let X be the general complete intersection threefold of type (2,4) or of type (3,3) in P5. There are infinitely many smooth rigid rational curves on X and these curves generate a subgroup of infinite rank in CH2(X).
Some recent work of C. Voisin gives a generalisation of Griffiths’ results for all threefolds with trivial canonical bundle. It would be interesting to see if the above result also holds in this generality.