Hilbert curves of 3-dimensional scrolls over surfaces

Let (X,L) be a 3-dimensional scroll over a smooth surface Y. Its Hilbert curve is an affine plane cubic consisting of a given line and a conic. This conic turns out to be the Hilbert curve of the Q-polarized surface (Y,12det⁡E), where E is the rank-2 vector bundle obtained by pushing down L via the scroll projection, if and only if E is properly semistable in the sense of Bogomolov.