Our main result is that the complexity of computing linear projections of an equidimensional, but non necessarily reduced, curve(or equivalently the degree-complexity of the Gröbner basis computation for elimination orders) has its maximal value, namely Bayer’s bound mo, if and only if the smallest linear subspace containingCis a plane. If this is so, mocoincides with the degree ofCand with the degree-complexity of the reverse lexicographic ordering.