Let $X$ be a normal Gorenstein complex projective variety. We introduce the Hilbert variety $V_X$ associated to the Hilbert polynomial $\chi(x_1 L_1,\ldots, x_\rho L_\rho)$, where $ L_1,\ldots, L_\rho$ is a basis of $\Pic X$, $\rho$ being the Picard number of $X$, and $x_1,\ldots,x_\rho$ are complex variables. After studying general properties of $V_X$ we specialize to the Hilbert curve of a polarized variety $(X,L)$, namely the plane curve of degree $\dim(X)$ associated to $\chi(xK_X+yL)$. Special emphasis is given to the case of polarized $3$-folds

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.

In this paper varieties are always assumed to be defined over the field [Copf ] of complex numbers.Given a smooth projective variety Z, the classification of smooth projective varieties X containing Z as an ample divisor occupies an extremely important position in the theory of polarized varieties and it is well-known that the structure of Z imposes severe restrictions on that of X. Inspired by this philosophy, we set up the following condition ([midast ]) in [LM1] in order to generalize several results on ample divisors to ample vector bundles:([midast ]) [Escr ] is an ample vector bundle of rank r [ges ] 2 on a smooth projective variety X of dimension n such that there exists a global section s ∈ Γ([Escr ]) whose zero locus Z = (s)0 is a smooth subvariety of X of dimension n − r [ges ] 1.

Let S be an ample vector bündle of rank n -2 > 2 on a complex projective manifold X of dimension n having a section whose zero locus is a smooth surface Z. Pairs (X, S) äs above are classified under the assumption that Z is a P 1 -bündle over a smooth curve. We also prove that ( ) = -oo if Z is a birationally ruled surface. 1991 Mathematics Subject Classification: 14J60; 14C20, 14F05, 14J26. IntroductionAt early 80's Bädescu studied ample divisors with special emphasis on projective space bundles over smooth curves. In particular, in a series of papers [Bl], [B2], [B 3], he determined which projective threefolds can contain any geometrically ruled surface äs an ample divisor. In fact äs was pointed out also in a subsequent paper where he extended bis result to normal varieties [B4, Theorem 7], the most difficult case is that of P 1 -bundles over curves [B4, Remark 2, p. 13]. According to the idea of revisiting the philosophy about ample divisors (e.g. see [BS2, Ch. 5]) in the more general setting of ample vector bundles äs expressed in [LM], here we consider the following set-up.(0.1) X is a complex projective manifold of dimension n and S is an ample vector bündle of rank n -2 > 2 on X such that there exists a section s e (

Let E be an ample vector bundle of rank n − 2 ≥ 2 on a complex projective manifold X of dimension n having a section whose zero locus is a smooth surface Z. We determine the structure of pairs (X, E) as above under the assumption that Z is a properly elliptic surface. This generalizes known results on threefolds containing an elliptic surface as a smooth ample divisor. Among the applications we prove a conjecture relating the Kodaira dimension of X to that of Z, and we show that if 0 ≤ κ(Z) ≤ 1, then p g (Z) > 0 unless X is a P n−2 -bundle over a smooth surface S with p g (S) = 0.Introduction.