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Globally Generated Vector Bundles with Small $c_{\tiny 1}$ on Projective Spaces
About this Title
Cristian Anghel, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO–014700, Bucharest, Romania, Iustin Coandă, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO–014700, Bucharest, Romania and Nicolae Manolache, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO–014700, Bucharest, Romania
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
MSC: Primary 14J60; Secondary 14H50, 14N25
Table of Contents
Chapters
- Introduction
- 1. Preliminaries
- 2. Some general results
- 3. The cases $c_1=4$ and $c_1 = 5$ on $\mathbb {P}^2$
- 4. The case $c_1 = 4$, $c_2 = 5, 6$ on $\mathbb {P}^3$
- 5. The case $c_1 = 4$, $c_2 = 7$ on $\mathbb {P}^3$
- 6. The case $c_1 = 4$, $c_2 = 8$ on $\mathbb {P}^3$
- 7. The case $c_1 = 4$, $5 \leq c_2 \leq 8$ on $\mathbb {P}^n$, $n \geq 4$
- A. The case $c_1 = 4$, $c_2 = 8$, $c_3 = 2$ on $\mathbb {P}^3$
- 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 $c_1 \leq 5$ on 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). We also propose a conjecture concerning the classification of globally generated vector bundles with $c_1 \leq n - 1$ on the projective $n$-space. We verify the conjecture for $n \leq 5$.- Hirotachi Abo, Wolfram Decker, and Nobuo Sasakura, An elliptic conic bundle in $\mathbf P^4$ arising from a stable rank-$3$ vector bundle, Math. Z. 229 (1998), no. 4, 725–741. MR 1664785, DOI 10.1007/PL00004679
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