On the minimum number of points covered by a set of lines in $PG(2, q)$

Abstract

Segre (Ann Mat Pura Appl 48:1–96, 1959) mentioned that the number $N$ of points on a curve which splits into $k$ distinct lines on the projective plane over a finite field of order $q$ satisfies $kq - \frac{k(k-3)}{2} \le N \le kq+1.$ We see that the upper bound is satisfactory, but the lower one is not for $k\ge q+2$ [resp. $k\ge q+3$] if $q$ is odd [resp. even]. We consider the minimum number $m_q(k)$ of points on such a curve of degree $k$, and obtain the complete sequence $\{m_q(k) \mid 0 \le k\le q^2+q+1\}$ for every prime power $q\le 8$.