The fixation line in the .-coalescent

We define a Markov process in a forward population model with backward genealogy given by the .-coalescent. This Markov process, called the fixation line, is related to the block counting process through its hitting times. Two applications are discussed. The probability that the n-coalescent is deeper than the (n.1)-coalescent is studied. The distribution of the number of blocks in the last coalescence of the n-Beta(2..,.)-coalescent is proved to converge as n.., and the generating function of the limiting random variable is computed.