Smooth, isolated curves in families of Calabi-Yau threefolds in homogeneous spaces (Q2861453)

This paper studies the embeddings of complete projective curves into Calabi-Yau complete intersection threefolds in homogeneous spaces. Special attention is paid to families of Calabi-Yau threefolds in \({\mathbb{P}}^m\), for \(7\leq m\leq 12\), that are complete intersections in certain homogeneous spaces. The main result is the existence of isolated curves of various degrees and genera in each of these six families of Calabi-Yau complete intersection threefolds.NEWLINENEWLINETheorem 1: For \(m=7\), the general Calabi-Yau threefold of type \((3,1,1)\cap G(2,V^5)\subseteq {\mathbb{P}}^7\) contains an isolated smooth irreducible curve of degree \(d\) and genus \(g\) if \(0\leq g\leq 4\) and \(d^2\geq 20g-4\) or \(5\leq g\leq 13\) and \(d\geq g+6\).NEWLINENEWLINEFor \(8\leq m\leq 12\), the general Calabi-Yau threefold contains an isolated smooth irreducible curve of degree \(d\) and genus \(g\) if \(0\leq g\leq m-4\) and \(d^2\geq 4(m-3)g-4\).NEWLINENEWLINEHere \(G(2,V^5)\) denotes the Grassmann variety of \(2\)-dimensional subspace of a vector space \(V\) of dimension \(5\).NEWLINENEWLINEThis theorem extends the author's earlier result [Trans. Am. Math. Soc. 364, No. 10, 5243--5264 (2012; Zbl 1330.14014)] to the case where the ambient space is not a projective space. The method used to obtain these results is more or less the same as the earlier paper.NEWLINENEWLINEAlong the way, the author classifies all degrees and genera of smooth curves on BN general \(K3\) surfaces of genus \(\mu\), where \(5\leq \mu\leq 10\). (For the definition of a BN general \(K3\) surface, see \textit{S. Mukai} [Sugaku Expo. 15, No. 2, 125--150 (2001); translation from Sugaku 47, No. 2, 125--144 (1995; Zbl 0889.14020)].)NEWLINENEWLINETheorem 2: Let \(4\leq n\leq 9,\, d>0\), \(g\geq 0\) be integers. Then there exists a BN general \(K3\) surface \(X\) of degree \(2n\) in \({\mathbb{P}}^{m+1}\) (and genus \(\mu=n+1\)) containing a smooth irreducible curve \(C\) of degree \(d\) and genus \(g\), if and only if one of the following cases occurs:NEWLINENEWLINE(I) \(g=d^2/4n+1\) and \(2n\) divides \(d\); (II) \(n=5\) and \(g=(d^2+4)/20\); (III) \(n=7\) and \(g=(d^2+3)/28\); (IV) \(g=d^2/4n\) and \(d\) is not divisible by \(2n\); (V) \(g<d^2/4n\) and \((d,g)\neq (2n+1, n+1).\)NEWLINENEWLINEFurthermore, in case (I) the surface \(X\) can be chosen such that \(\mathrm{Pic}X={\mathbb{Z}}[H]\) and in cases (II)-(V) such that \(Pic X={\mathbb{Z}}[H]\oplus {\mathbb{Z}}[C]\), where \(H\) is the hyperplane section of \(X\).