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Rational curves on a complete intersection Calabi-Yau variety in \({\mathbb{P}}^3\times{\mathbb{P}}^3\). (English) Zbl 1030.14016

Summary: We study rational curves on the Tian-Yau complete intersection Calabi-Yau threefold (CICY) in \(\mathbb{P}^3\times\mathbb{P}^3\). Existence of positive dimensional families of nonsingular rational curves is proved for every degree \(\geq 4\). 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.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14J81 Relationships between surfaces, higher-dimensional varieties, and physics
14M20 Rational and unirational varieties
14M10 Complete intersections
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