### Abstract

We study projective surfaces X ⊂ ℙ^{r} (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 ∩ ℙ^{r}^{–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) ⊂ ℙ^{5}, or else admit a plane ð�”½ = ℙ^{2} ⊂ ℙ^{r} 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 language | English |
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Pages (from-to) | 549-567 |

Number of pages | 19 |

Journal | Taiwanese Journal of Mathematics |

Volume | 21 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Taiwanese Journal of Mathematics*,

*21*(3), 549-567. https://doi.org/10.11650/tjm/7753