  Graber, Harris and Starr proved, when n >= 2d, the irreducibility of the
Hurwitz space H^0_{d,n}(Y) which parametrizes degree d coverings of a smooth,
projective curve Y of positive genus, simply branched in n points, with full
monodromy group S_d (math.AG/0205056). We sharpen this result and prove that
H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} and in the case of elliptic Y
if n >= max{2,2d-6}. We extend the result to coverings simply branched in all
but one point of the discriminant. Fixing the ramification multiplicities over
the special point we prove that the corresponding Hurwitz space is irreducible
if the number of simply branched points is >= 2d-2. We study also simply
branched coverings with monodromy group different from S_d and when n is large
enough determine the corresponding connected components of H_{d,n}(Y). Our
results are based on explicit calculation of the braid moves associated with
the standard generators of the n-strand braid group of Y.
