We study rational curves on
the Tian-Yau complete intersection Calabi–Yau threefold (CICY) in ℙ3× ℙ3.
Existence of positive dimensional families of nonsingular rational curves is
proved for every degree ≥ 4. The number of nonsingular rational curves of
degree 1,2,3 on a general Tian–Yau CICY is finite and enumerated. The
number of curves of these degrees are also enumerated for the special Tian–Yau
CICY. There are two 1-dimensional families of singular rational curves of
degree 3 on a general Tian–Yau CICY, making this degree a turning point
between finite and infinite number of curves. We also introduce a notion of
equivalence of a family of rational curves, and determine the equivalences of the
two 1-dimensional families on the Tian–Yau CICY. The equivalences equal
the predicted numbers of curves obtained by a power series expansion of
the solution of a Picard-Fuchs equation that arises in superconformal field
theory.
