  We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree
$d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it
is a classical result that for larger $d$ there are only cones). We apply this
to the study of the extension theory of pluricanonical curves and genus $3$
curves, whenever they verify Property $N_2$, using and slightly expanding the
theory of integration of ribbons of the authors and E.~Sernesi. We compute the
corank of the relevant Gaussian maps, and we show that all ribbons over such
curves are integrable, and thus there exists a universal extension.
  We carry out a similar program for linearly normal hyperelliptic curves of
degree $d\geq 2g+3$. We classify surfaces having such a curve $C$ as a
hyperplane section, compute the corank of the relevant Gaussian maps, and prove
that all ribbons over $C$ are integrable if and only if $d=2g+3$. In the latter
case we obtain the existence of a universal extension.
