By Conference on Algebraic Geometry (1988 Sundance Institute), Brian Harbourne, Robert Speiser
This quantity includes the lawsuits of the NSF-CBMS local convention on Algebraic Geometry, held in Sundance, Utah, in July 1988. The convention keen on algebraic curves and similar kinds. a number of the papers accumulated right here signify lectures introduced on the convention, a few document on learn performed throughout the convention, whereas others describe comparable paintings conducted in different places
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Additional info for Algebraic Geometry: Sundance 1988 : Proceedings of a Conference on Algebraic Geometry Held July 18-23, 1988 With Support from Brigham Young Universi
3. Historical remarks. The correspondence between locally complete intersections of codimension 2 in Pn and holomorphic 2bundles over Pn is essentially due to Serre . In its present form it has been repeatedly discovered (Horrocks , Barth–Van de Ven , Hartshorne , Grauert–M¨ ulich ). Here we have followed more or less the presentation of Hartshorne. We remark that this correspondence also holds for manifolds; the attentive reader will easily discover which conditions must be required.
Let E = ((Λn−2 TPn (−1)) ⊗ OPn (−1))∗ . The sequence which is dual to (*) is ⊕(n+1) 0 → TPn (−2) → OPn 2 → E → 0, which shows that E is globally generated of rank n+1 n r= −n= . 2 2 For n ≥ 3 we have r ≥ n; thus there is an exact sequence ⊕(r−n) 0 → OPn →E→E →0 with a holomorphic n-bundle E . E is again globally generated. The top Chern class of E is cn (E ) = cn (E) = 0. Thus E contains a trivial subbundle of rank 1. , 0 → OPn → E → F → 0 is exact. Thus we have found an (n − 1)-bundle F over Pn with 1 1 − 2h c(F ) = c(E ) = c(E) = = .
1. Determine the largest integer k = k(n) such that uniform k-bundles over Pn are homogeneous. 38 1. HOLOMORPHIC VECTOR BUNDLES AND THE GEOMETRY OF Pn From the examples of Hirschowitz we have in any case k(n) ≤ 3n−2 for n ≥ 3. , uniform n-bundles over Pn are always of the form OPn (a1 ) ⊕ · · · ⊕ OPn (an ) or TPn (a) or Ω1Pn (a). Furthermore we conjecture that uniform (n + 1)-bundles over Pn are of the form OPn (a1 ) ⊕ · · · ⊕ OPn (an+1 ) or TPn (a) ⊕ OPn (b) or Ω1Pn (a) ⊕ OPn (b). This would give k(n) ≥ n + 1.
Algebraic Geometry: Sundance 1988 : Proceedings of a Conference on Algebraic Geometry Held July 18-23, 1988 With Support from Brigham Young Universi by Conference on Algebraic Geometry (1988 Sundance Institute), Brian Harbourne, Robert Speiser