The authors provide a complete classification of globally generated vector bundles with first Chern class $c_1 \leq 5$ one the projective plane and with $c_1 \leq 4$ on the projective $n$-space for $n \geq 3$. This reproves and extends, in a systematic manner, previous results obtained for $c_1 \leq 2$ by Sierra and Ugaglia [ J. Pure Appl. Algebra 213 (2009), 2141-2146], and for $c_1 = 3$ by Anghel and Manolache [ Math. Nachr. 286 (2013), 1407-1423] and, independently, by Sierra and Ugaglia [ J. Pure Appl. Algebra 218 (2014), 174-180]. It turns out that the case $c_1 = 4$ 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). The authors also propose a conjecture concerning the classification of globally generated vector bundles with $c_1 \leq n - 1$ on the projective $n$-space. They verify the conjecture for $n \leq 5$.
Introduction
Acknowledgements
Preliminaries
Some general results
The cases $c_1=4$ and $c_1 = 5$ on ${\mathbb P}^2$
The case $c_1 = 4$, $c_2 = 5, 6$ on ${\mathbb P}^3$
The case $c_1 = 4$, $c_2 = 7$ on ${\mathbb P}^3$
The case $c_1 = 4$, $c_2 = 8$ on ${\mathbb P}^3$
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 ${\mathbb P}^3$
Bibliography.
Cristian Anghel, The Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania.
Iustin Coanda, The Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania.
Nicolae Manolache, The Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania.
