Projective surfaces of maximal sectional regularity

TL;DR

This paper classifies projective surfaces of maximal sectional regularity in projective space, showing they are either cones over maximal regularity curves or projections of rational scrolls, and analyzes their cohomological properties.

Contribution

It provides a complete classification of such surfaces, establishes the regularity equality, and computes their cohomological invariants and extremal varieties.

Findings

Surfaces are either cones over maximal regularity curves or projections of rational scrolls.

The regularity of these surfaces equals d - r + 3.

Extremal variety is either a plane or a rational scroll S(1,1,1) in P^5.

Abstract

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 $\reg(C)$ 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 $\widetilde{X} \subset \mathbb{P}^{d+1}$. 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 $S(1,1,1) \subset \mathbb{P}^5$.