Projective surfaces of maximal sectional regularity

Abstract

We study projective surfaces $X\subset {\mathbb{P}}^r$ (with $r\ge 5$) of maximal sectional regularity and degree $d>r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(C)$ of a general hyperplane section curve $C=X\cap {\mathbb{P}}^{r-1}$ takes the maximally possible value $d-r+3$. We show that each of these surfaces is either a cone over a curve $C\subset {\mathbb{P}}^{r-1}$ of maximal regularity or else a birational outer linear projection of a smooth rational surface scroll $\tilde{X}\subset {\mathbb{P}}^{d+1}$. We prove that the Castelnuovo-Mumford regularity of these surfaces satisfies the equality $\reg(X) = d-r+3$ and we compute or estimate various of their cohomological invariants as well as their Betti numbers. We study the the extremal variety $\mathbb{F}(X)$ of these surfaces $X$, that is the closed union of the extremal secant lines of all smooth hyperplane section curves of $X$. We show that $\mathbb{F}(X)$ is either a plane or that otherwise $r=5$ and $\mathbb{F}(X)$ is a rational smooth threefold scroll $S(1,1,1)\subset {\mathbb{P}}^5$.

54 pages