### 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 |
---|---|

Pages (from-to) | 549-567 |

Number of pages | 19 |

Journal | Taiwanese Journal of Mathematics |

Volume | 21 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Taiwanese Journal of Mathematics*,

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

**On surfaces of maximal sectional regularity.** / Brodmann, Markus; Lee, Wanseok; Park, Euisung; Schenzel, Peter.

Research output: Contribution to journal › Article

*Taiwanese Journal of Mathematics*, vol. 21, no. 3, pp. 549-567. https://doi.org/10.11650/tjm/7753

}

TY - JOUR

T1 - On surfaces of maximal sectional regularity

AU - Brodmann, Markus

AU - Lee, Wanseok

AU - Park, Euisung

AU - Schenzel, Peter

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Castelnuovo-Mumford regularity

KW - Extremal

KW - Variety of maximal sectional regularity

UR - http://www.scopus.com/inward/record.url?scp=85019756441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019756441&partnerID=8YFLogxK

U2 - 10.11650/tjm/7753

DO - 10.11650/tjm/7753

M3 - Article

AN - SCOPUS:85019756441

VL - 21

SP - 549

EP - 567

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

SN - 1027-5487

IS - 3

ER -