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## SOME COMBINATORICS OF BINOMIAL COEFFICIENTS AND THE BLOCH-GIESEKER PROPERTY FOR SOME HOMOGENEOUS BUNDLES (2001)

### Citations

96 |
Tangents and Secants of Algebraic Varieties
- Zak
- 1993
(Show Context)
Citation Context ...e existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], [H2], [H3], [Ho], [HM], [KPR], [LV], [OSS], [T], [V], =-=[Z]-=-). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 on P5) [Ho], are the null-correlation bundles [OSS], ... |

53 |
Varieties of small codimension in projective space,
- Hartshorne
- 1974
(Show Context)
Citation Context ...1)(1+4t)( n p3)(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], [H2], =-=[H3]-=-, [Ho], [HM], [KPR], [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 on P5)... |

37 |
Monads and moduli of vector bundles
- Barth, Hulek
- 1978
(Show Context)
Citation Context ...2)(1+(p+1)t) (1+2t) ( np1)(1+4t)( n p3)(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians (=-=[BH]-=-, [E1], [EHS], [H2], [H3], [Ho], [HM], [KPR], [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks... |

34 |
Positive polynomials for ample vector bundles,
- Fulton, Lazarsfeld
- 1983
(Show Context)
Citation Context ... classes are trivial and the other Chern classes are positive We note that in the theorem above, case (i) gives the null-correlation bundle, and case (ii) gives Tango's example. Fulton and Lazarsfeld =-=[FL]-=- generalized Bloch and Gieseker's result and showed that the Schur polynomials for ample bundles are positive. So the next question is Question. Are the Schur polynomials forspPn(p+ 1) positive? S. Ka... |

30 |
A rank 2 vector bundle on P4 with 15,000 symmetries,
- Horrocks, Mumford
- 1973
(Show Context)
Citation Context ...p3)(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], [H2], [H3], [Ho], =-=[HM]-=-, [KPR], [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 on P5) [Ho], are t... |

20 |
The positivity of the Chern classes of an ample vector bundle,
- Bloch, Gieseker, et al.
- 1971
(Show Context)
Citation Context ...p + 1) is generated by global sections and its Chern classes are nonnegative, this is the same question as Is cn p Pn(p+ 1) positive? A theorem that immediately comes to mind is by Bloch and Gieseker =-=[BG]-=-. If E is an ample vector bundle of rank r on an n-dimensional variety, and if r > n, then all Chern classes of E are numerically positive. So we give the following Denition. A vector bundle has the ... |

13 |
Algebraic vector bundles on projective spaces: A problem list, Topology 18
- Hartshorne
- 1979
(Show Context)
Citation Context ... ( np1)(1+4t)( n p3)(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], =-=[H2]-=-, [H3], [Ho], [HM], [KPR], [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 ... |

8 |
An Example of Indecomposable Vector Bundle of Rank n−1
- Tango
- 1976
(Show Context)
Citation Context ... even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], [H2], [H3], [Ho], [HM], [KPR], [LV], [OSS], =-=[T]-=-, [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 on P5) [Ho], are the null-correlation bundl... |

8 |
Constructing vector bundles from codimension-two subvarieties, PhD Thesis. Gianfranco Casnati, Dipartimento di Scienze Matematiche, Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129
- Vogelaar
(Show Context)
Citation Context ...). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], [H2], [H3], [Ho], [HM], [KPR], [LV], [OSS], [T], =-=[V]-=-, [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 on P5) [Ho], are the null-correlation bundles [O... |

7 |
Generalized null correlation bundles
- Ein
- 1988
(Show Context)
Citation Context ...p+1)t) (1+2t) ( np1)(1+4t)( n p3)(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], =-=[E1]-=-, [EHS], [H2], [H3], [Ho], [HM], [KPR], [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundl... |

6 |
Construction of low rank vector bundles on
- Kumar, Peterson, et al.
(Show Context)
Citation Context ...(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], [H2], [H3], [Ho], [HM], =-=[KPR]-=-, [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 on P5) [Ho], are the null... |

1 |
Some remarks on Buchsbaum bundles
- Chang
(Show Context)
Citation Context ... Tango's example is very simple: for a globally generated bundle of rank r n, and with trivial top Chern class, its quotient by r n+ 1 sections is a vector bundle. This same idea was also used in =-=[C]-=- to characterize Buchsbaum bundles. Along these lines, a natural question to ask is DoesspPn(p+ 1) have trivial top Chern class? SincespPn(p + 1) is generated by global sections and its Chern classes ... |

1 |
Private communication
- Ein
(Show Context)
Citation Context ...ynomials forspPn(p+ 1) positive? S. Katz and S. Stromme have written computer software [KS] to compute the numerical invariants we are interested for given projective spaces. We also note that L. Ein =-=[E2]-=- has a nice argument to show the vanishing of cn n2Pn (n 1) (which is the same as the Segre class ofsPn(2)) by looking at the map to the Grassmannian of lines given by the tautological line bundle.... |

1 |
uniformes de rang au plus n sur Pn(C) sont ceux qu'on croit
- Elencwajg, Hirschowitz, et al.
- 1979
(Show Context)
Citation Context ... (1+2t) ( np1)(1+4t)( n p3)(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], =-=[EHS]-=-, [H2], [H3], [Ho], [HM], [KPR], [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of r... |

1 |
Algebraic geometry, GTM 152
- Hartshorne
- 1977
(Show Context)
Citation Context ...stimate the coecients of the Chern polynomial ofspPn(p + 1). Recall that the Chern polynomial is the polynomial whose coecient of tk is the kth Chern class. It is multiplicative for exact sequences =-=[H1]-=-. We use the Koszul resolution ofspPn(p + 1) to write the Chern polynomial of pPn(p + 1) as the quotient of the products of Chern polynomials (1 + at) N of NLOPn(a). Hence the problem is reduced to de... |

1 |
Examples of rank three vector bundles on projective space
- Horrocks
- 1978
(Show Context)
Citation Context ...t)( n p3)(1+pt)( n 1) (for p even). The existence (or nonexistence) of low rank vector bundles on the projective n-space has been intriguing to many mathematicians ([BH], [E1], [EHS], [H2], [H3], =-=[Ho]-=-, [HM], [KPR], [LV], [OSS], [T], [V], [Z]). For n 4, the only known bundles of rank r < n, besides the Horrocks-Mumford bundle (of rank 2 on P4) [HM], and the Horrocks bundle (of rank 3 on P5) [Ho],... |

1 |
A Maple package for intersection theory
- Katz, Stromme
(Show Context)
Citation Context ...ed that the Schur polynomials for ample bundles are positive. So the next question is Question. Are the Schur polynomials forspPn(p+ 1) positive? S. Katz and S. Stromme have written computer software =-=[KS]-=- to compute the numerical invariants we are interested for given projective spaces. We also note that L. Ein [E2] has a nice argument to show the vanishing of cn n2Pn (n 1) (which is the same as th... |