Abstract

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.

Authors