On globally generated vector bundles on projective spaces

Abstract

a b s t r a c tA classification is given for globally generated vector bundles E of rank k on P n having first Chern class c 1 (E) = 2. In particular, we get that they split if k < n unless E is a twisted null-correlation bundle on P 3 . In view of the well-known correspondence between globally generated vector bundles and maps to Grassmannians, we obtain, as a corollary, a classification of double Veronese embeddings of P n into a Grassmannian G(k − 1, N) of (k − 1)-planes in P N .

On globally generated vector bundles on projective spaces II

“…Remark 3. As in [12], one can easily deduce the classification of triple Veronese embeddings of P r in a Grassmannian of (k − 1)-planes from Theorem 1.1 and Corollary 1.2. The case k = 2 has been studied in [8].…”

“…Globally generated vector bundles with c 1 ≤ 2 were classified in [12]. From now on, we concentrate on the case c 1 = 3.…”

“…We would like to thank Edoardo Ballico for pointing out the following gap in the proof of [12,Proposition 3.2].…”

Geometry of the Sasakura bundle

“…This last section aims to describe the place of the Sasakura bundle in this context, following the joint paper with Coanda and Manolache [2]. Chronologically, the main contributions in this area are: -Chiodera and Ellia [8] in ('12) determined the Chern classes of rank 2 globally generated vector bundles with c 1 ≤ 5 on P n , -Sierra and Ugaglia [14] in ('09) classified globally generated vector bundles with c 1 ≤ 2 on P n , -Sierra and Ugaglia [15] in ('14) and independently Manolache with the author [3] in ('13) classified globally generated vector bundles with c 1 ≤ 3 on P n , -Coanda, Manolache and the author [2] ('13) classified globally generated vector bundles with c 1 ≤ 4 on P n .…”

On the singular scheme of split foliations

“…In [28] J. C. Sierra and L. Ugaglia proved that a globally generated vector bundle F on P n such that rank(F ) < n and c 1 (F ) = 2, always splits unless F is a twisted null-correlation bundle on P 3 . Using this we can prove the following.…”