Open Access
February, 2020 The Hilbert Curve of a $4$-dimensional Scroll with a Divisorial Fiber
Antonio Lanteri, Andrea Luigi Tironi
Taiwanese J. Math. 24(1): 31-41 (February, 2020). DOI: 10.11650/tjm/190206


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.


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Antonio Lanteri. Andrea Luigi Tironi. "The Hilbert Curve of a $4$-dimensional Scroll with a Divisorial Fiber." Taiwanese J. Math. 24 (1) 31 - 41, February, 2020.


Received: 27 July 2018; Revised: 27 January 2019; Accepted: 25 February 2019; Published: February, 2020
First available in Project Euclid: 12 March 2019

MathSciNet: MR4053836
zbMATH: 07175538
Digital Object Identifier: 10.11650/tjm/190206

Primary: 14C20 , 14N30
Secondary: 14J35 , 14M99

Keywords: divisorial fiber , Hilbert curve , scroll

Rights: Copyright © 2020 The Mathematical Society of the Republic of China

Vol.24 • No. 1 • February, 2020
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