  We give a bound on the Castelnuovo-Mumford regularity of a homogeneous ideals
I, in a polynomial ring A, in terms of number of variables and the degrees of
generators, when the dimension of A/I is at most two. This bound improves the
one obtained by Caviglia and Sbarra in this case. In the continuation of the
examples constructed by Clare D'Cruz and the first author, we use families of
monomial curves to construct homogeneous ideals showing that these bounds are
quite sharp.
