On Q-split Bézout intersection

AbstractThis paper deals with the following question: Can one find a linear pencil of plane curves of a given degree m, defined over the rational number field Q, with m2 distinct Q-rational base points, such that every curve belonging to the pencil is irreducible? The answer for m = 3 is well known, which gives an elliptic curve defined over Q(t) with Mordell-Weil rank r = 8. For general m, an affirmative answer will give an algebraic curve over Q(t) with rank r = m2 − 1 (cf) [7]. The case m = 4 is solved in the affirmative in this paper. The question is open for m >4