Projective Curves with maximal regularity and applications to syzygies and surfaces

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.