Elliptic surfaces and ample vector bundles

Abstract

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.

Connections between the geometry of a projective variety and of an ample section

“…This claim can be proved exactly as in the last part of the proof of [3, Theorem 1.4]. [17] showed that the elliptic fibration Φ| Z : Z → Y has actually no multiple fibers and the genus of the curve Y is g(Y ) = h 1,0 (Z).…”

“…If moreover r = 1 the effective cones of the general fibers of X and Z coincide, so this can be viewed as a relative version of the results in Section 3. Finally we apply our results to the case in which Z is a surface with Kodaira dimension 0 or 1, not necessarily minimal, giving a different proof of some of the results obtained in [16] and [17].…”

“…If dim Z = 2 we can study the case 0 ≤ κ(Z) < dim Z = 2 without the assumption of minimality; this was done in [16] and [17], here we give a different proof. In higher dimensions, even for dim Z = 3, this is much more difficult.…”

On the extendability of elliptic surfaces of rank two and higher

“…-The proof of the extension of π under hypothesis (ii) is inspired by [32]. In case X is smooth it has been proved first by Ionescu [21] (see also [12], [14]).…”

Ample vector bundles and Bordiga surfaces

“…(10) there is a vector bundle V on a smooth curve B such that X = P(V) and E F ∼ = O P (1) ⊕(n−2) for every fiber F ∼ = P n−1 of the projection π : X → B; (11) there is a vector bundle V on a smooth curve B such that X = P(V) and E F ∼ = O P (2) ⊕ O P (1) ⊕(n−3) for every fiber F ∼ = P n−1 of the projection π : X → B; (12) there is a surjective morphism ψ : X → B onto a smooth curve B such that any general fiber F of ψ is isomorphic to a smooth quadric Q n−1 ⊂ P n , with E F ∼ = O Q (1) ⊕(n−2) ; (13) there is a vector bundle U on a smooth surface W such that X = P(U) and E F ∼ = O P (1) ⊕(n−2) for every fiber F ∼ = P n−2 of the projection π : X → W . Now suppose that the ample vector bundle E of rank n − 2 admits a section vanishing on a smooth surface Z.…”