Globally generated vector bundles of rank 2 on a smooth quadric threefold

Abstract

Abstract. We investigate the existence of globally generated vector bundles of rank 2 with c1 ≤ 3 on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on Q with c1 = 3 is an odd instanton up to twist.

Globally generated vector bundles on the Segre threefold with Picard number two

“…In particular π 2| C is always an embedding and so deg(J ∩ C) ≤ 1 for each line J ⊂ X of bidegree (1, 0). Fix any smooth and connected C ⊂ X with bidegree (3,3). Since h 0 (O C (3, 3)) = 6 by the Rieman-Roch theorem, we have h 0 (I C (1, 2)) ≥ 3.…”

“…If they are nontrivial they must have strictly positive first Chern class. The classification of globally generated vector bundles with low first Chern class has been done over several rational varieties such as projective spaces [1,13] and quadric hypersurfaces [3]. There is also a recent work over complete intersection Calabi-Yau threefolds by the authors [4].…”

“…Lemma 4. 3 gives that E has rank 2 and that there are P ∈ P 1 and a line L ⊂ P 2 such that C = {P } × L. Since h 0 (I C (1, 0)) = 1 and h 0 (I C (−1, 1)) = h 0 (I C (0, 0)) = 0, there is a non-zero map f : O X (1, 1) − → E with coker(f ) torsion free, i.e. coker(f ) ∼ = I T (1, 0) with either T = ∅ or T a locally complete intersection curve.…”

On globally generated vector bundles on projective spaces II

“…Following the research initiated in [12], globally generated vector bundles and reflexive sheaves with low first Chern class on projective spaces and quadric hypersurfaces have recently been studied by several authors (see [5,6,10,[1][2][3][4]). …”

Globally generated vector bundles on a smooth quadric surface

“…We ask similar questions over a smooth quadric surface Q and give answers as in our previous works [4] [5]. Our main theorem is the following: (1, 1, 2; r = 2, 3), (1, 2, 2; 2), (1, 2, 3; r = 2, 3), (1,2,4; r = 2, 3, 4, 5), (2, 2, 3; 2), (2,2,4; r = 2, 3), (2, 2, 5; r = 2, 3), (2,2,6; r = 2, 3, 4, 5), (2, 2, 8; r = 2, 3, 4, 5, 6, 7, 8) In the second section we fix the notations and we explain the preliminaries. In the third section we show that any globally generated vector bundle with c 1 = (0, 0) or c 1 = (a, 0) and either a ≤ 2 or rank two splits.…”