Rational points on hyperelliptic curves having a marked non-Weierstrass point

In this paper, we consider the family of hyperelliptic curves over Q having a fixed genus n and a marked rational non-Weierstrass point. We show that when n >= 9, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as n tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of 5/2 on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137-1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and L-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23-91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.