The irreducibility and monodromy of some families of linear series with imposed ramifications

Suppose that the adjusted Brill-Noether number is zero, we prove that there exists a family of twice-marked smooth projective curves such that the family of linear series with two imposed ramification conditions is irreducible. Moreover, under certain conditions, we show that the monodromy group contains the alternating group. In the case $r=1$, the monodromy group is the full symmetric group.