On surfaces of maximal sectional regularity

Markus Brodmann, Wanseok Lee, Euisung Park, Peter Schenzel

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1 Citation (Scopus)

Abstract

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 use the classi_cation of varieties of maximal sectional regularity of [5] to see that these surfaces are either particular divisors on a smooth rational 3-fold scroll S(1; 1; 1) ⊂ ℙ5, or else admit a plane ð�”½ = ℙ2 ⊂ ℙr such that (Formula Presented) 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 cohomological invariants as well as the Betti numbers of such surfaces.

Original languageEnglish
Pages (from-to)549-567
Number of pages19
JournalTaiwanese Journal of Mathematics
Volume21
Issue number3
DOIs
Publication statusPublished - 2017

Keywords

  • Castelnuovo-Mumford regularity
  • Extremal
  • Variety of maximal sectional regularity

ASJC Scopus subject areas

  • Mathematics(all)

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