On surfaces of maximal sectional regularity

We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d > r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(\mathcal{C})$ of a general hyperplane section curve $\mathcal{C} = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We use the classification of varieties of maximal sectional regularity of \cite{BLPS1} to see that these surfaces are either particular divisors on a smooth rational $3$-fold scroll $S(1,1,1)\subset \mathbb{P}^5$, or else admit a plane $\mathbb{F} = \mathbb{P}^2 \subset \mathbb{P}^r$ such that $X \cap \mathbb{F} \subset \mathbb{F}$ is a pure curve of degree $d-r+3$. We show that our surfaces are either cones over curves of maximal regularity, or almost non-singular projections of smooth rational surface scrolls. We use this to show that the Castelnuovo-Mumford regularity of such a surface $X$ satisfies the equality $\reg(X) = d-r+3$ and we compute or estimate various of the cohomological invariants as well as the Betti numbers of such surfaces. We also study the geometry of extremal secant lines of our surfaces $X$, more precisely the closure $\Sigma(X)$ of the set of all proper extremal secant lines to $X$ in the Grassmannian $\mathbb{G}(1, \mathbb{P}^r).$