  We compute the class of the locus in M_g of curves having a pencil with two
unspecified triple ramification points. This is the first example of a
geometric divisor on M_g which is not the pull-back of a divisor on the moduli
space of pseudo-stable curves. This space, in which elliptic tails are replaced
by cusps, appears as a result of the first divisorial contraction in the
minimal model program for M_g. In particular, we show that our divisor picks-up
the locus of Fermat cubic tails when restricted to the boundary divisor of
elliptic tails. We also give various enumerative applications concerning
coverings of the generic curve having special ramification behaviour.
