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., Advance publication (2019), 11 pages.

First available in Project Euclid: 12 March 2019

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Digital Object 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., advance publication, 12 March 2019. doi:10.11650/tjm/190206. https://projecteuclid.org/euclid.twjm/1552377618

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