Taiwanese Journal of Mathematics

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

Antonio Lanteri and Andrea Luigi Tironi

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

scroll divisorial fiber Hilbert curve


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

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