Taiwanese Journal of Mathematics

The Hilbert Curve of a $4$-dimensional Scroll with a Divisorial Fiber

Antonio Lanteri and Andrea Luigi Tironi

Full-text: Open access

Abstract

In dimension $n = 2m-2 \geq 4$ adjunction theoretic scrolls over a smooth $m$-fold may not be classical scrolls, due to the existence of divisorial fibers. A $4$-dimensional scroll $(X,L)$ over $\mathbb{P}^3$ of this type is considered, and the equation of its Hilbert curve $\Gamma$ is determined in two ways, one of which relies on the fact that $(X,L)$ is at the same time a classical scroll over a threefold $Y \neq \mathbb{P}^3$. It turns out that $\Gamma$ does not perceive divisorial fibers. The equation we obtain also shows that a question raised in [2] has negative answer in general for non-classical scrolls over a $3$-fold. More precisely, the answer for $(X,L)$ is negative or positive according to whether $(X,L)$ is regarded as an adjunction theoretic scroll or as a classical scroll; in other words, it is the answer to this question to distinguish between the existence of jumping fibers or not.

Article information

Source
Taiwanese J. Math., Volume 24, Number 1 (2020), 31-41.

Dates
Received: 27 July 2018
Revised: 27 January 2019
Accepted: 25 February 2019
First available in Project Euclid: 12 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1552377618

Digital Object Identifier
doi:10.11650/tjm/190206

Mathematical Reviews number (MathSciNet)
MR4053836

Zentralblatt MATH identifier
07175538

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves 14N30: Adjunction problems
Secondary: 14J35: $4$-folds 14M99: None of the above, but in this section

Keywords
scroll divisorial fiber Hilbert curve

Citation

Lanteri, Antonio; Tironi, Andrea Luigi. The Hilbert Curve of a $4$-dimensional Scroll with a Divisorial Fiber. Taiwanese J. Math. 24 (2020), no. 1, 31--41. doi:10.11650/tjm/190206. https://projecteuclid.org/euclid.twjm/1552377618


Export citation

References

  • M. C. Beltrametti, A. Lanteri and M. Lavaggi, Hilbert surfaces of bipolarized varieties, Rev. Roumaine Math. Pures Appl. 60 (2015), no. 3, 281–319.
  • M. C. Beltrametti, A. Lanteri and A. J. Sommese, Hilbert curves of polarized varieties, J. Pure Appl. Algebra 214 (2010), no. 4, 461–479.
  • M. C. Beltrametti and A. J. Sommese, Comparing the classical and the adjunction-theoretic definition of scrolls, in: Geometry of Complex Projective Varieties, 55–74, Sem. Conf. 9, Mediterranean, Rende, 1993.
  • ––––, The Adjunction Theory of Complex Projective Varieties, De Gruyter Expositions in Mathematics 16, Walter de Gruyter & Co., Berlin, 1995.
  • P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, 1978.
  • A. Lanteri, Characterizing scrolls via the Hilbert curve, Internat. J. Math. 25 (2014), no. 11, 1450101, 17 pp.
  • A. Lanteri and A. L. Tironi, Hilbert curve characterizations of some relevant polarized manifolds, arXiv:1803.01131.
  • A. L. Tironi, Scrolls over four dimensional varieties, Adv. Geom. 10 (2010), no. 1, 145–159.
  • ––––, Nefness of adjoint bundles for ample vector bundles of corank $3$, Math. Nachr. 286 (2013), no. 14-15, 1548–1570.