On the Castelnuovo-Mumford regularity of ideals, in dimension 2
  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. 
Marc Chardin, Amadou Lamine Fall