Taiwanese Journal of Mathematics

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

This article is in its final form and can be cited using the date of online publication and the DOI.

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., Advance publication (2019), 11 pages.

Dates
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

Citation

Lanteri, Antonio; Tironi, Andrea Luigi. The Hilbert Curve of a $4$-dimensional Scroll with a Divisorial Fiber. Taiwanese J. Math., advance publication, 12 March 2019. doi:10.11650/tjm/190206. https://projecteuclid.org/euclid.twjm/1552377618

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.