Ample vector bundles with zero loci of sectional genus two

Abstract

Let X be a smooth complex projective variety and let Z ⊂ X be a smooth submanifold of dimension 2, which is the zero locus of a section of an ample vector bundle E of rank dim X − dim Z 2 on X. Let H be an ample line bundle on X, whose restriction H Z to Z is generated by global sections. Triplets (X, E, H ) as above are classified under the assumption that (Z, H Z ) is a polarized manifold of sectional genus 2. This can be regarded as a step towards the classification of ample vector bundles of corank one and curve genus two.

Ample vector bundles with zero loci of small Δ-genera

“…If n − r = 1, then H is any ample line bundle on X except in Case (10). In that case, let ξ be the tautological line bundle of V, let δ be a line bundle on B and let G be a rank-(n − 1) vector bundle on B such that E ∼ = ξ ⊗ π * G; then H = αξ + π * δ where the integer α satisfies the condition α deg V + α deg G + deg δ ≥ 3.…”

“…The following argument is inspired by [10]. We can write X ∼ = P P 1 (V), where V ∼ = n i=1 O P 1 (a i ), with a 1 ≥ a 2 ≥ · · · ≥ a n = 0.…”

Ample vector bundles and Bordiga surfaces

“…Hence α = 2, which gives H = 2ξ + F ; moreover s = 4, which shows that (Z, H Z ) must be a Bordiga surface of degree 12. By arguing as in [7,Lemma 1.2] we can see that E fits into an exact sequence…”

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

“…Indeed, triplets (X, E, H ) with g(Z , H Z ) = 3 are classified and investigated in [9] under the assumption that |H | defines an embedding of Z (we note that triplets (X, E, H ) with g(Z , H Z ) = 2 are classified in [3] when H Z is generated by its global sections). Indeed, triplets (X, E, H ) with g(Z , H Z ) = 3 are classified and investigated in [9] under the assumption that |H | defines an embedding of Z (we note that triplets (X, E, H ) with g(Z , H Z ) = 2 are classified in [3] when H Z is generated by its global sections).…”

“…Then (X, E, H ) = (P 3 Let E be an ample vector bundle of rank r 2 on a smooth projective variety X of dimension n such that there exists a global section s of E whose zero locus Z = (s) 0 is a smooth subvariety of dimension n − r 3 of X .…”