We study projective surfaces X⊂Pr (with r≥5) of maximal sectional regularity and degree d>r, hence surfaces for which the Castelnuovo-Mumford regularity
Nearby in the stack
(C)
of a general hyperplane section curve
C=X∩Pr−1
takes the maximally possible value
d−r+3
. We show that each of these surfaces is either a cone over a curve
C⊂Pr−1
of maximal regularity or else a birational outer linear projection of a smooth rational surface scroll
X⊂Pd+1
. We prove that the Castelnuovo-Mumford regularity of these surfaces satisfies the equality
(X) = d-r+3
and we compute or estimate various of their cohomological invariants as well as their Betti numbers. We study the the extremal variety
F(X)
of these surfaces
X
, that is the closed union of the extremal secant lines of all smooth hyperplane section curves of