On the Intersections of Rational Curves with Cubic Plane Curves

Let V be a generic quintic threefold in the 4-dimensional complex projective space P. A well-known conjecture of Clemens says that V has only a finite number of rational curves in each degree. Although Clemens’ conjecture is still quite open at this moment (it is known to be true for degree up to 7 by S. Katz [K]), recently physicists, based on the mirror symmetry principle, have proposed a formula to predict the numbers of rational curves of various degrees on V which is a Calabi-Yau threefold [M]. Furthermore, some of these predicted numbers have been verified mathematically. Now if S is a generic quartic surface inP, then we know that for each positive integer k, there are only a finite number of rational curves in H(S,O(k)) by the classification theory of surfaces. In this case, the number of hyperplane sections with 3 nodes is known classically to be 3200 (cf. [V], [YZ]). The main result of this paper is the following