Vojta's Inequality and Rational and Integral Points of Bounded Degree on Curves

Let C in C_1xC_2 be a curve of type (d_1,d_2) in the product of the two curves C_1 and C_2. Let d be a positive integer. We prove that if a certain inequality involving d_1, d_2, d, and the genera of the curves C_1, C_2, and C is satisfied, then the set of points P in C(\kbar) with [k(P):k]<=d is finite for any number field k. We prove a similar result for integral points of bounded degree on C. These results are obtained as consequences of an inequality of Vojta which generalizes the Roth-Wirsing theorem to curves.