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2017 On Surfaces of Maximal Sectional Regularity
Markus Brodmann, Wanseok Lee, Euisung Park, Peter Schenzel
Taiwanese J. Math. 21(3): 549-567 (2017). DOI: 10.11650/tjm/7753

Abstract

We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d \gt r$, hence surfaces for which the Castelnuovo-Mumford regularity $\operatorname{reg}(\mathcal{C})$ of a general hyperplane section curve $\mathcal{C} = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We use the classification of varieties of maximal sectional regularity of [5] to see that these surfaces are either particular divisors on a smooth rational $3$-fold scroll $S(1,1,1) \subset \mathbb{P}^5$, or else admit a plane $\mathbb{F} = \mathbb{P}^2 \subset \mathbb{P}^r$ such that $X \cap \mathbb{F} \subset \mathbb{F}$ 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 $\operatorname{reg}(X) = d-r+3$ and we compute or estimate various cohomological invariants as well as the Betti numbers of such surfaces.

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Markus Brodmann. Wanseok Lee. Euisung Park. Peter Schenzel. "On Surfaces of Maximal Sectional Regularity." Taiwanese J. Math. 21 (3) 549 - 567, 2017. https://doi.org/10.11650/tjm/7753

Information

Published: 2017
First available in Project Euclid: 1 July 2017

zbMATH: 06871331
MathSciNet: MR3661380
Digital Object Identifier: 10.11650/tjm/7753

Subjects:
Primary: 13D02, 14H45

Rights: Copyright © 2017 The Mathematical Society of the Republic of China

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Vol.21 • No. 3 • 2017
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