Ample vector bundles of small curve genera

Abstract

Let e be a vector bundle of rank n À 1 on a smooth complex projective variety X of dimension n^3, and let gXY e be the curve genus of XY e defined by the formula 2gXY e À 2 K X c 1 ec nÀ1 e, where K X is the canonical bundle of X. Then it is proved that gXY e is a nonnegative integer if e is ample. Moreover, polarized pairs XY e with gXY e % 1 are completely classified.Introduction. In this paper varieties are always assumed to be defined over the field C of complex numbers.Let X be a smooth projective variety of dimension n^2, and let e be a vector bundle of rank n À 1 on X. We define the curve genus gXY e of XY e by the formula

Ample vector bundles with zero loci of small Δ-genera

“…By [9, (6.3)], ∆(Z, H Z ) = 1 implies that (Z, H Z ) is a del Pezzo manifold; then g(X, F) = 1, where F = E ⊕ H ⊕(n−r−1) . So we are in the assumption of [25,Theorem 2] which gives the following possibilities for (X, F):…”

“…On the other hand we should note that for n − r = 1 if H Z is very ample and non-special, then ∆(Z, H Z ) is simply the genus g of the smooth curve Z. Thus our problem overlaps that of classifying pairs (X, E) as in 1.1 with E having curve genus g. As far as we know, results on this related problem are available only for g ≤ 2, with E being very ample when equality holds [25], [26]. For n − r ≥ 2, starting from the known classification of projective manifolds of small ∆ and using miscellaneous results concerning ample vector bundles, we get satisfactory structure theorems for our triplets (X, E, H).…”

Very ample vector bundles of curve genus two

“…case (ii)). However, we should note that g(X, E ) is an integer without assuming the condition ( * ), and that the pairs (X, E ) with g(X, E ) Ϲ 1 have been classified in [7] when E is simply supposed to be ample.…”

Remarks on ample vector bundles of curve genus two

“…Therefore we assume that n ≥ 3. Then it follows from [4,Proposition] that the curve genus g(X, E) of (X, E) is an integer. Furthermore, if E is ample, then g(X, E) ≥ 0, and the pairs (X, E) with g(X, E) ≤ 1 are classified in [4, Theorems 1 and 2].…”