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

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14H10 Families, moduli of curves (algebraic)
##### Keywords:
quintic threefold; Dwork pencil; line
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