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

In dimension n = 2m – 2 ≥ 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 ℙ³ of this type is considered, and the equation of its Hilbert curve Γ 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 ≠ ℙ³. It turns out that Γ 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.