Very ample vector bundles of curve genus two

Abstract

Let E be a very ample vector bundle of rank n − 1 on a smooth complex projective variety X of dimension n м 3, and let g(X, E ) be the curve genus of (X, E ) defined by the formula 2g(X,Introduction. In what follows, varieties are always assumed to be defined over the field C of complex numbers.Given a smooth projective curve C, the classification of smooth projective surfaces X containing C as a very ample, or even merely ample, divisor is an important problem in the theory of polarized surfaces. In order to generalize several results on very ample divisors on smooth projective surfaces to very ample vector bundles on smooth projective varieties X of dimension n м 2, it is natural to consider very ample vector bundles E of rank n − 1 on X, which satisfy the following condition ( * ):

Notes on very ample vector bundles on 3-folds

“…Let us compute c 1 (E) 3 and c 1 (E)c 2 (E). First, by (2.3) and (2.4), Especially for 3-folds, Theorem 5 allows us to improve [MS,Theorem 6] as follows:…”

Ample vector bundles with zero loci of small Δ-genera

“…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).…”

Ample vector bundles with zero loci of sectional genus two

“…The classification of pairs (X, F) as above is known only for g(X, F) 1 [15]. For g(X, F) = 2 a classification result is provided in [16] under the stronger assumption that the vector bundle F is very ample. Now, let E be an ample vector bundle on X of rank r n − 2, and suppose that ( * ) E admits a section vanishing on a smooth submanifold Z ⊂ X of the expected codimension r.…”

“…Let X be a smooth projective variety of dimension n endowed with an ample vector bundle F of rank n−1 2, with g(X, F) = 2. According to [16,Theorem 6,case (2)] one of the possibilities for (X, F) is the following.…”

“…In particular F itself cannot be very ample. At least when B = P 1 we can add the following remark to the main result of [16]. In the same setting, when g(B) = 1 we can only assert that X = P(V), where V is a normalized indecomposable vector bundle of rank n on B of degree deg V 2.…”