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A nonsingular surface of degree $d \geq 2$ in $\mathbb{P}^3$ over$\mathbb{F}_q$ has at most $((d-1)q+1)d$ $\mathbb{F}_q$-lines, and this boundis optimal for $d = 2, \sqrt{q}+1, q+1$.

A bound for the number of lines lying on a nonsingular surface in $3$-space over a finite field