Sommervoll, Dag Einar Rational curves on a complete intersection Calabi-Yau variety in \({\mathbb{P}}^3\times{\mathbb{P}}^3\). (English) Zbl 1030.14016 Pac. J. Math. 192, No. 2, 415-430 (2000). 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. Cited in 1 Document 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 Keywords:supersymmetric theories for a 10-dimensional universe; Tian-Yau complete intersection Calabi-Yau threefold; number of nonsingular rational curves; Picard-Fuchs equation PDFBibTeX XMLCite \textit{D. E. Sommervoll}, Pac. J. Math. 192, No. 2, 415--430 (2000; Zbl 1030.14016) Full Text: DOI