Calabi-Yau threefolds with \(\rho > 13\)
If a Calabi-Yau threefold \(X\) has Picard number \(\rho(X)>13\), then there is a birational morphism \(\varphi:X\to\overline X\) onto a normal projective threefold \(\overline X\) with \(\rho(\overline X)<\rho(X)\) and for which the exceptional locus of \(\varphi\) is covered by rational curves. This result improves one in an earlier paper of the second named author [Invent. Math. 98, No. 1, 139-155 (1989; Zbl 0688.14032)], where a similar result was proved under the assumption \(\rho(X)>19\). As in the earlier paper, the proof consists of combining classification methods from algebraic geometry with arguments of Davenport concerning diophantine approximation for integral cubic forms, the cubic form in question being the cup-product on the cohomology group \(H^ 2(X,\mathbb{Z})\). The improvement occurs because it is not actually necessary to search for integral zeros of the cubic form as was done in the previous paper, but instead for integral points at which the form is negative but with sufficiently small absolute value. This also has implications concerning the cone of Kähler classes in \(H^ 2(X,\mathbb{R})\), and its intersection being sparse in the cubic cone defined by the cup-product.
diophantine approximation for integral cubic forms
scientific article; zbMATH DE number 126355