A bound for the number of lines lying on a nonsingular surface in   $3$-space over a finite field
  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 bound is optimal for $d = 2, \sqrt{q}+1, q+1$. 
Masaaki Homma and Seon Jeong Kim
math.AG