Globally generated vector bundles on a smooth quadric surface

Abstract

Abstract. We give the classification of globally generated vector bundles of rank 2 on a smooth quadric surface with c 1 ≤ (2, 2) in terms of the indices of the bundles, and extend the result to arbitrary higher rank case. We also investigate their indecomposability and give the sufficient and necessary condition on numeric data of vector bundles for indecomposability.

Quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$

“…Now if E fits into sequence (2.1), then by Lemma 3.2 (2, 0) is an index if and only if (0, 2) is also an index, and this, again from [BHM15], is equivalent to an extension of type…”

“…Indecomposable globally generated vector bundles with low first Chern class on a smooth quadric surface have been classified in the paper [BHM15]. The authors prove that there exist such indecomposable and globally generated vector bundles of rank 2 on Q with c 1 = (2, 2) if and only if c 2 = 3, 4, 5, 6, 8.…”

“…One of the tools used in [BHM15] is the notion of index: a pair (p, q) ∈ Z 2 is an index for a globally generated vector bundle E on Q if it is a maximal pair such that the twist E(−p, −q) has global sections: H 0 (E(−p, −q)) = 0. Here one considers (p, q) (p ′ , q ′ ) if and only if p p ′ and q q ′ .…”

“…Proof. Analyzing the classification from [BHM15] in light of Proposition 2.1, Lemma 3.2, and Proposition 3.3, we are able to rule out a few cases, and are left with the ones listed above.…”

“…However, since rank 2 globally generated bundles on a smooth quadric surface are classified (see [BHM15]), we have considered the following problems: first, understanding which of these vector bundles are associated to a quadratic system of skewsymmetric matrices of constant rank 4; second, studying the geometry of the found examples, relating them to linear congruences of lines in P 5 .…”