Read e-book online Algebraic Geometry: Sundance 1988 : Proceedings of a PDF

By Conference on Algebraic Geometry (1988 Sundance Institute), Brian Harbourne, Robert Speiser

ISBN-10: 0821851241

ISBN-13: 9780821851241

ISBN-10: 1719842272

ISBN-13: 9781719842273

ISBN-10: 6119904174

ISBN-13: 9786119904170

ISBN-10: 7418727267

ISBN-13: 9787418727262

ISBN-10: 9919874094

ISBN-13: 9789919874094

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

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3. Historical remarks. The correspondence between locally complete intersections of codimension 2 in Pn and holomorphic 2bundles over Pn is essentially due to Serre [112]. In its present form it has been repeatedly discovered (Horrocks [66], Barth–Van de Ven [9], Hartshorne [55], Grauert–M¨ ulich [45]). 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.

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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


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