  We define a Markov process in a forward population model with backward
genealogy given by the $\Lambda$-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 $n$-$\operatorname
{Beta}(2-\alpha,\alpha)$-coalescent is proved to converge as
$n\rightarrow\infty$, and the generating function of the limiting random
variable is computed.
