2013/05/10 by Markus Brodmann, Wanseok Lee, Brodmann, Markus +5
Computer Science · Mathematics · #13D02 #14H45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1305.2355
openalex publication_date 2013/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \reg(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 \widetildeX ⊂ ℙd+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 \mathbbF(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 \mathbbF(X) is either a plane or that otherwise r =5 and \mathbbF(X) is a rational smooth threefold scroll S(1,1,1) ⊂ ℙ5.