# The Hilbert curve of a 4-dimensional scroll with a divisorial fiber

## TL;DR

This paper investigates the Hilbert curve of a 4-dimensional scroll with divisorial fibers over a threefold, revealing that the curve does not detect divisorial fibers and addressing a question about classical versus non-classical scrolls.

## Contribution

It determines the Hilbert curve of a specific 4-dimensional scroll and shows it does not detect divisorial fibers, clarifying the distinction between classical and non-classical scrolls.

## Key findings

- Hilbert curve equation derived in two ways
- Curve does not perceive divisorial fibers
- Negative answer to a previous question for non-classical scrolls

## 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 $Y \not=\mathbb P^3$. It turns out that $\Gamma$ does not perceive divisorial fibers. The equation we obtain also shows that a question raised in a previous article by Beltrametti, Lanteri and Sommese, 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.