Characterizing some polarized Fano fibrations via Hilbert curves

The Hilbert curve of a complex polarized manifold (X,L) is the complex
affine plane curve of degree dim(X) defined by the Hilbert-like polynomial
chi(xK_X+yL), where K_X is the canonical bundle of X and x and y are regarded
as complex variables. A natural expectation is that this curve encodes several
properties of the pair (X,L). In particular, the existence of a fibration of X over a
variety of smaller dimension induced by a suitable adjoint bundle to L translates into
the fact that the Hilbert curve has a quite special shape. Along this line, Hilbert curves
of special varieties like Fano manifolds with low coindex, as well as fibrations over low
dimensional varieties having such a manifold as general fiber, endowed with
appropriate polarizations, are investigated. In particular, several polarized manifolds
relevant for adjunction theory are completely characterized in terms of their Hilbert curves.