## Taiwanese Journal of Mathematics

### On Surfaces of Maximal Sectional Regularity

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 21, Number 3 (2017), 549-567.

Dates
First available in Project Euclid: 1 July 2017

https://projecteuclid.org/euclid.twjm/1498874606

Digital Object Identifier
doi:10.11650/tjm/7753

Mathematical Reviews number (MathSciNet)
MR3661380

Zentralblatt MATH identifier
06871331

#### Citation

Brodmann, Markus; Lee, Wanseok; Park, Euisung; Schenzel, Peter. On Surfaces of Maximal Sectional Regularity. Taiwanese J. Math. 21 (2017), no. 3, 549--567. doi:10.11650/tjm/7753. https://projecteuclid.org/euclid.twjm/1498874606

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