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...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], ...

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...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)...

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...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...

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... 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...

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...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...

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...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 ...

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... ( 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 ...

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... 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...

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...). 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...

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...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...

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...(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...

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... 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 ...

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...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....

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... (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...

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...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...

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...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],...

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...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...