  We first show that the union of a projective curve with one of its extremal
secant lines satisfies the linear general position principle for hyperplane
sections. We use this to give an improved approximation of the Betti numbers of
curves ${\mathcal C} \subset \mathbb P^r_K$ of maximal regularity with $\deg
{\mathcal C} \leq 2r -3.$ In particular we specify the number and degrees of
generators of the vanishing ideal of such curves. We apply these results to
study surfaces $X \subset \mathbb P^r_K$ whose generic hyperplane section is a
curve of maximal regularity. We first give a criterion for "an early decent of
the Hartshorne-Rao function" of such surfaces. We use this criterion to give a
lower bound on the degree for a class of these surfaces. Then, we study
surfaces $X \subset \mathbb P^r_K$ for which $h^1(\mathbb P^r_K, {\mathcal
I}_X(1))$ takes a value close to the possible maximum $\deg X - r +1.$ We give
a lower bound on the degree of such surfaces. We illustrate our results by a
number of examples, computed by means of {\sc Singular}, which show a rich
variety of occuring phenomena.
