Characterizing scrolls via the Hilbert curve

Abstract

Let X be a projective bundle over a smooth curve and let L be an ample line bundle on X inducing [Formula: see text] on every fiber. The Hilbert curve Γ 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 Γ 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.