Construction of low rank vector bundles on $\mathbf {P}^{4}$ and $\mathbf {P}^{5}$

Authors:
N. Mohan Kumar, Chris Peterson and A. Prabhakar Rao

Journal:
J. Algebraic Geom. **11** (2002), 203-217

DOI:
https://doi.org/10.1090/S1056-3911-01-00309-5

Published electronically:
November 20, 2001

MathSciNet review:
1874112

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Abstract |
References |
Additional Information

Abstract: We describe a technique which permits a uniform construction of a number of low rank bundles, both known and new. In characteristic two, we obtain rank two bundles on $\mathbf {P}^{5}$. In characteristic $p$, we obtain rank two bundles on $\mathbf {P}^4$ and rank three bundles on $\mathbf {P}^5$. In arbitrary characteristic, we obtain rank three bundles on $\mathbf {P}^4$ and rank two bundles on the quadric $S_5$ in $\mathbf {P}^6$.

Sas H. Abo, W. Decker and N. Sasakura, *An elliptic conic bundle in $\mathbf {P}^4$ arising from a stable rank-3 vector bundle*, Mathematische Zeitschrift **229** no.4 (1998), 725–741.
Ancona V. Ancona and G. Ottaviani, *The Horrocks bundles of rank three on ${P}^ 5$*, J. Reine Angew. Math. **460** (1995), 69–92.
EG G. Evans and P. Griffith, *Syzygies*, London Mathematical Society Lecture Note Series Vol. 106, Cambridge University Press, 1985.
HM G. Horrocks and D. Mumford, *A rank 2 vector bundle on $\mathbf {P}^4$ with 15,000 symmetries*, Topology **12** (1973), 63–81.
Horrocks G. Horrocks, *Vector bundles on the punctured spectrum of a local ring II*, Vector Bundles on Algebraic Varieties (Bombay, 1984), TIFR, Oxford University Press (1987) 207–216.
Horrocks2 G. Horrocks, *Examples of rank three vector bundles on five-dimensional projective space*, Journal of the London Mathematical Society **18** (1978), 15–27.
Mohan2 N. Mohan Kumar, *Construction of rank two vector bundles on $\mathbf {P}^4$ in positive characteristic*, Inventiones Mathematicae **130** (1997), 277–286.
Tango1 H. Tango, *An example of indecomposable vector bundle of rank $n-1$ on $\mathbf {P}^4$*, Journal of Mathematics of Kyoto University **16** no. 1 (1976), 137–141.
Tango2 H. Tango, *On morphisms from projective space $\mathbf {P}^n$ to the Grassmann variety $\mathbf {Gr}(n, d)$*, Journal of Mathematics of Kyoto University **16** no. 1 (1976), 201–207.
Vett U. Vetter, *Zu einem Satz von G. Trautmann über den Rang gewisser kohärenter analytischer Moduln*, Archiv der Mathematik (Basel) **24** (1973), 158–161.

Sas H. Abo, W. Decker and N. Sasakura, *An elliptic conic bundle in $\mathbf {P}^4$ arising from a stable rank-3 vector bundle*, Mathematische Zeitschrift **229** no.4 (1998), 725–741.
Ancona V. Ancona and G. Ottaviani, *The Horrocks bundles of rank three on ${P}^ 5$*, J. Reine Angew. Math. **460** (1995), 69–92.
EG G. Evans and P. Griffith, *Syzygies*, London Mathematical Society Lecture Note Series Vol. 106, Cambridge University Press, 1985.
HM G. Horrocks and D. Mumford, *A rank 2 vector bundle on $\mathbf {P}^4$ with 15,000 symmetries*, Topology **12** (1973), 63–81.
Horrocks G. Horrocks, *Vector bundles on the punctured spectrum of a local ring II*, Vector Bundles on Algebraic Varieties (Bombay, 1984), TIFR, Oxford University Press (1987) 207–216.
Horrocks2 G. Horrocks, *Examples of rank three vector bundles on five-dimensional projective space*, Journal of the London Mathematical Society **18** (1978), 15–27.
Mohan2 N. Mohan Kumar, *Construction of rank two vector bundles on $\mathbf {P}^4$ in positive characteristic*, Inventiones Mathematicae **130** (1997), 277–286.
Tango1 H. Tango, *An example of indecomposable vector bundle of rank $n-1$ on $\mathbf {P}^4$*, Journal of Mathematics of Kyoto University **16** no. 1 (1976), 137–141.
Tango2 H. Tango, *On morphisms from projective space $\mathbf {P}^n$ to the Grassmann variety $\mathbf {Gr}(n, d)$*, Journal of Mathematics of Kyoto University **16** no. 1 (1976), 201–207.
Vett U. Vetter, *Zu einem Satz von G. Trautmann über den Rang gewisser kohärenter analytischer Moduln*, Archiv der Mathematik (Basel) **24** (1973), 158–161.

Additional Information

**N. Mohan Kumar**

Affiliation:
Department of Mathematics, Washington University, Saint Louis, Missouri 63130

Email:
kumar@math.wustl.edu

**Chris Peterson**

Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523

MR Author ID:
359254

Email:
peterson@math.colostate.edu

**A. Prabhakar Rao**

Affiliation:
Department of Mathematics, University of Missouri - Saint Louis, Saint Louis, Missouri 63121

Email:
rao@arch.umsl.edu

Received by editor(s):
May 11, 2000

Published electronically:
November 20, 2001