Lines on the Dwork pencil of quintic threefolds.

*(English)*Zbl 1279.14050This 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 \(\psi=\infty,\, \varphi^2=-3/4\).

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 \(\psi=\infty,\, \varphi^2=-3/4\).

Reviewer: Noriko Yui (Kingston)