Characterizing scrolls via the Hilbert curve

Let X be a projective bundle over a smooth curve and let L be an ample line bundle on X inducing O_{\mathbb P}(r) on every fiber. The Hilbert curve \Gamma of such a polarized manifold (X,L) is explicitly described and the reconstruction problem is addressed. In particular, it is shown that the very special geographic shape of \Gamma allows to recover the adjunction theoretic type of (X,L) for smooth surfaces, for scrolls over a smooth curve, as well as for Veronese bundles under additional assumptions.