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Globally Generated Vector Bundles with Small $c_\tiny 1$ on Projective Spaces


About this Title

Cristian Anghel, Iustin Coandă and Nicolae Manolache

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 253, Number 1209
ISBNs: 978-1-4704-2838-9 (print); 978-1-4704-4413-6 (online)
DOI: https://doi.org/10.1090/memo/1209
Published electronically: March 29, 2018
Keywords:projective space, vector bundle, globally generated sheaf

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Table of Contents


Chapters

  • Introduction
  • Chapter 1. Preliminaries
  • Chapter 2. Some general results
  • Chapter 3. The cases $c_1=4$ and $c_1 = 5$ on $\mathbb P^2$
  • Chapter 4. The case $c_1 = 4$, $c_2 = 5, 6$ on $\mathbb P^3$
  • Chapter 5. The case $c_1 = 4$, $c_2 = 7$ on $\mathbb P^3$
  • Chapter 6. The case $c_1 = 4$, $c_2 = 8$ on $\mathbb P^3$
  • Chapter 7. The case $c_1 = 4$, $5 \leq c_2 \leq 8$ on $\mathbb P^n$, $n \geq 4$
  • Appendix A. The case $c_1 = 4$, $c_2 = 8$, $c_3 = 2$ on $\mathbb P^3$
  • Appendix B. The case $c_1 = 4$, $c_2 = 8$, $c_3 = 4$ on $\mathbb P^3$

Abstract


We provide a complete classification of globally generated vector bundles with first Chern class on the projective plane and with on the projective -space for . This reproves and extends, in a systematic manner, previous results obtained for by Sierra and Ugaglia [J. Pure Appl. Algebra 213(2009), 2141âĂŞ2146], and for by Anghel and Manolache [Math. Nachr. 286(2013), 1407âĂŞ1423] and, independently, by Sierra and Ugaglia [J. Pure Appl. Algebra 218(2014), 174âĂŞ180]. It turns out that the case is much more involved than the previous cases, especially on the projective 3-space. Among the bundles appearing in our classification one can find the Sasakura rank 3 vector bundle on the projective 4-space (conveniently twisted). We also propose a conjecture concerning the classification of globally generated vector bundles with on the projective -space. We verify the conjecture for .

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