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arXiv · 2609.26862

A smooth quintic surface with 85 lines and Picard number 43

Xun Yu·Zigang Zhu
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Abstract

We construct a pencil of complex quintic surfaces with every smooth member containing exactly $85$ lines. We show that one smooth member has Picard number $43$ and by introducing the notion of a $d$-saturated lattice, we prove that its Néron--Severi group is generated by the lines and four conics. We also construct smooth quintic surfaces with $79$ lines, including examples with $24$ pairwise skew lines, and a smooth surface of degree $10$ with $356$ lines, including $112$ pairwise skew lines.

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Xun Yu, Zigang Zhu. 2026-09-22. A smooth quintic surface with 85 lines and Picard number 43. https://arxiv.org/abs/2609.26862

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