Narrow Results

Let (X, L) be a polarized manifold of dimension 3. In this paper, we consider a lower bound for h⁰(K_X + 2L). We prove that h⁰(K_X + 2L) > 0 if K_X + 2L is nef, which is a conjecture of Beltrametti–Sommese for polarized 3-folds. Moreover we classify polarized 3-folds (X, L) with h⁰(K_X + 2L) = 1 under the assumption that K_X + 2L is nef.

Let X be a projective bundle over a smooth curve and let L be an ample line bundle on X inducing 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.

Let $(X,L)$ be a quadric fibration over a smooth curve. The explicit equation of the corresponding Hilbert curve $\mathrm{\Gamma }$ is obtained. The geometry of $\mathrm{\Gamma }$ reflects some structure properties of $(X,L)$; in particular, its special shape allows us to recognize that $(X,L)$ is a quadric fibration. In fact $\mathrm{\Gamma }$ is reducible into $dimX-2$ parallel lines with prescribed slope, evenly spaced, plus a conic. On the other hand, this conic can itself be regarded as the Hilbert curve of a polarized surface only in very rare circumstances.

Let $(X,L)$ be any Fano manifold polarized by a positive multiple of its fundamental divisor $H$. The polynomial defining the Hilbert curve of $(X,L)$ reduces to the Hilbert polynomial of $(X,H)$, hence it is totally reducible over $ℂ$; moreover, some of the linear factors appearing in the factorization have rational coefficients, e.g. if $X$ has index $\ge 2$. It is natural to ask when the same happens for all linear factors. Here the total reducibility over $\mathbb{Q}$ of the Hilbert polynomial is investigated for three special kinds of Fano manifolds: Fano manifolds of large index, toric Fano manifolds of low dimension, and projectivized Fano bundles of low coindex.

Let $P_{\lambda {\mathrm{\Sigma }}_n}$ be the Ehrhart polynomial associated to an integral multiple $\lambda$ of the standard simplex ${\mathrm{\Sigma }}_n\subset {ℝ}^n$. In this paper, we prove that if $(M,L)$ is an $n$-dimensional polarized toric manifold with associated Delzant polytope $\mathrm{\Delta }$ and Ehrhart polynomial $P_{\mathrm{\Delta }}$ such that $P_{\mathrm{\Delta }}=P_{\lambda {\mathrm{\Sigma }}_n}$, for some $\lambda \in {\mathbb{Z}}^{+}$, then $(M,L)\cong (ℂP^n,O(\lambda ))$ (where $O(1)$ is the hyperplane bundle on $ℂP^n$) in the following three cases: (1) arbitrary $n$ and $\lambda =1$, (2) $n=2$ and $\lambda =3$ and (3) $\lambda =n+1$ under the assumption that the polarization $L$ is asymptotically Chow semistable.

We find natural numbers m such that the dimensions of global sections of multiple adjoint bundles h⁰(m(K_X + L)) are strictly greater than zero for any quasi-polarized n-folds (X, L) for which X is a complex normal Gorenstein projective variety of dimension n with only rational singularities and K_X + L is nef.

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 $\chi (x{K}_X+yL)$, 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.