On the $k$-regularity of some projective manifolds

The conjecture on the (\textit{degree - codimension} + 1) - regularity of projective varieties is proved for smooth linearly normal polarized varieties ($X,L$) with $L$ very ample, for low values of $\Delta(X,L)$ =\textit{degree - codimension} - 1. Results concerning the projective normality of some classes of special varieties including scrolls over curves of genus 2 and quadric fibrations over elliptic curves, are proved.