zbMATH — the first resource for mathematics

Lines on the Dwork pencil of quintic threefolds. (English) Zbl 1279.14050
This paper is concerned with the lines in the Dwork pencil of quintic threefolds. These quintic threefolds \(M_{\psi}\) are defined as hypersurfaces in \(\mathbb{P}^4\) by \(\sum_{j=1}^5 x_j^5-5\psi\prod_{j=1}^5 x_j=0\) where \(\psi\) is a parameter. A quintic threefold \(M_{\psi}\) has \(101\) deformation parameters, and for generic values of these paramters, it has been observed that there are \(2875\) lines on \(M_{\psi}\). When \(\psi=0\), the Fermat quintic is known to have \(375\) lines, called the isolated lines. Van Geemen noticed in the 1990s that there are \(5000\) special lines in the \(M_{\psi}\).
A. Mustaţǎ [Math. Ann. 355, No. 1, 97–130 (2013; Zbl 1266.14032)] has proved that \(M_{\psi}\) with \(\psi\neq 0\) contains two continuous families of lines, parametrized by isomorphic curves \(\tilde{C}_{\pm \varphi}\), of genus \(626\) with Euler number \(-1250\). This gives, together with the \(375\) isolated lines, \(375+2\times 1250=2875\) lines on \(M_{\psi}\).
This paper studies in detail the curves \(\tilde{C}_{\pm \varphi}\) which parametrizes lines. These curves are \(125: 1\) covers of genus \(6\) curves \(C_{\pm \varphi}\). The curves \(C_{\pm \varphi}\) are presented as curves in \(\mathbb{P}^1\times \mathbb{P}^1\) that have \(3\) nodes. The blowing-up of \(\mathbb{P}^1\times \mathbb{P}^1\) at three notes is the del Pezzo surface \(dP_5\) of degree \(5\), whose automorphism group is the symmetric group \(S_5\) of degree \(5\) (which is also the symmetry group of the pair of curves \(C_{\pm\varphi}\)). Odd permutations exchange \(C_{\varphi}\) with \(C_{-\varphi}\) while even permutations preserve each curve. There are \(10\) exceptional curves on \(dP_5\) each intersects the curve \(C_{\varphi}\) in two points corresponding to van Geemen lines. The curves \(C_{\varphi}\) are shown in fact the curves of the Wiman pencil.
The family of lines are also studied when the Dwork pencil become singular, e.g., the cases \(\psi^5=1,\, \varphi^2=125/4\) and \(\psi=\infty,\, \varphi^2=-3/4\).

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
Full Text: DOI arXiv