A smooth quintic surface with 85 lines and Picard number 43

Source 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.