On Higher Rank Globally Generated Vector Bundles over a Smooth Quadric Threefold

Abstract

We give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold with c1 ≤ 2 and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated indecomposable vector bundles, and give the sufficient and necessary conditions on numeric data of vector bundles for indecomposability.Theorem 1.1. Let E be a non-split vector bundle of rank 3 on Q with c 1 ≤ 2. E is globally generated if and only if E admits an exact sequence,

Globally generated vector bundles on P1×P1×PBallico1, Huh2, Malaspina3 2016Journal of AlgebraSelf Cite1020View full textAdd to dashboardCite

“…In particular two of the integers e 1 , e 2 , e 3 are ones. Hence if deg(C) = 3, then C has multidegree (1, 1, 1), while if deg(C) = 4, then C has multidegree either (2, 1, 1), (1, 2, 1) or (1,1,2). By symmetry one of them occurs if and only if all the three possibilities occur, but they give different families of bundles.…”

“…) for every integer m with 2 ≤ m ≤ r 0 − 1. The Hartshorne-Serre correspondence gives the existence of a globally generated vector bundle E with Y as a dependency locus and no trivial factor (see [2,Lemma 4.1]) for all ranks r with 3 ≤ r ≤ r 0 .…”

“…, up to permutations on (e 1 , e 2 , e 3 ); C is linearly normal in its linear span and E is as in Remark 3.3 (2) and (3) with h 0 (E) = r + 2. In case (i) for each 3 ≤ r ≤ 7 the bundles are parametrized by an irreducible family.…”

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

“…In [8] Chiodera and Ellia classify the globally generated vector bundles of rank 2 on P n with small first Chern classes. 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.…”