A property of Hilbert curves of scrolls over surfaces

Let (X,L) be a polarized manifold of dimension n. Its Hilbert curve is an affine algebraic plane curve of degree n encoding properties related to fibrations of X, defined by suitable adjoint linear systems to L. In particular, if (X,L) is a scroll over a smooth surface S, its Hilbert curve consists of n−2 parallel lines with a given slope and evenly spaced, plus a conic. Making its equation explicit, we show that this conic turns out to be itself the Hilbert curve of the ℚ-polarized surface (Formula presented.), where ℰ is the rank-(n−1) vector bundle obtained by pushing down L via the scroll projection, if and only if ℰ is properly semistable in the sense of Bogomolov.