We consider the facial structure of the Uncapacitated Lot-sizing problem with start-up costs. We show that under mild assumptions on the objective function cost structure, adding valid inequalities described by Wolsey (1989) to the linear programming relaxation of the problem guarantees optimal integer solutions. For the special cases of a time horizon of 3 or less, we show that the inequalities completely describe the convex hull of feasible integer solutions. We also identify a new class of valid inequalities for the problem.