Taiwanese Journal of Mathematics

On Surfaces of Maximal Sectional Regularity

Markus Brodmann, Wanseok Lee, Euisung Park, and Peter Schenzel

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874606

Digital Object Identifier
doi:10.11650/tjm/7753

Mathematical Reviews number (MathSciNet)
MR3661380

Zentralblatt MATH identifier
06871331

Subjects
Primary: 14H45: Special curves and curves of low genus 13D02: Syzygies, resolutions, complexes

Keywords
Castelnuovo-Mumford regularity variety of maximal sectional regularity extremal locus extremal variety

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


Export citation

References

  • J. Ahn and S. Kwak, Graded mapping cone theorem, multisecants and syzygies, J. Algebra 331 (2011), no. 1, 243–262.
  • C. Albertini and M. Brodmann, A bound on certain local cohomology modules and application to ample divisors, Nagoya Math. J. 163 (2001), 87–106.
  • M.-A. Bertin, On the regularity of varieties having an extremal secant line, J. Reine Angew. Math. 545 (2002), 167–181.
  • ––––, On singular varieties having an extremal secant line, Comm. Algebra 34 (2006), no. 3, 893–909.
  • M. Brodmann, W. Lee, E. Park and P. Schenzel, On projective varieties of maximal sectional regularity, J. Pure Appl. Algebra 221 (2017), no. 1, 98–118.
  • M. Brodmann and P. Schenzel, On projective curves of maximal regularity, Math. Z. 244 (2003), no. 2, 271–289.
  • ––––, Projective curves with maximal regularity and applications to syzygies and surfaces, Manuscripta Math. 135 (2011), no. 3-4, 469–495.
  • M. Chardin, A. L. Fall and U. Nagel, Bounds for the Castelnuovo-Mumford regularity of modules, Math. Z. 258 (2008), no. 1, 69–80.
  • M. Decker, G. M. Greuel and H. Schönemann, Singular $3-1-2$ – A computer algebra system for polynomial computations, (2011). http://www.singular.uni-kl.de
  • D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133.
  • T. Fujita, Classification Theories of Polarized Varieties, London Mathematical Society Lecture Note Series 155, Cambridge University Press, Cambridge, 1990.
  • L. Gruson, R. Lazarsfeld and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491–506.
  • S. Kwak and E. Park, Some effects of property $N_p$ on the higher normality and defining equations of nonlinearly normal varieties, J. Reine Angew. Math. 582 (2005), 87–105.
  • R. Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423–429.
  • A. Noma, Multisecant lines to projective varieties, in Projective Varieties with Unexpected Properties, A volume in memory of Giuseppe Veronese, Proceedings of the international conference `Varieties with Unexpected Properties', Siena, Italy, June 8–13, 2004. Edited by Ciliberto, Ciro et al., Walter de Gruyter, Berlin, (2005), 349–359.
  • E. Park, On syzygies of divisors on rational normal scrolls, Math. Nachr. 287 (2014), no. 11-12, 1383–1393.
  • H. C. Pinkham, A Castelnuovo bound for smooth surfaces, Invent. Math. 83 (1986), no. 2, 321–332.