New PDF release: Algebraic Topology, Gottingen 1984

By L. Smith

ISBN-10: 0387160612

ISBN-13: 9780387160610

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3 we have fµα |Uαβ = fµβ |Uαβ over intersections and therefore there are global isomorphism fµ,α Fµ ∼ = Fµ′ , commuting with the surjections. This 38 proves q = q ′ . In order to verify (S2 ) let (Uα ) be an open cover of U and let qα = [EUα → F•α ] ∈ F (Uα ) be given with qα |Uαβ = qβ |Uαβ . As before this means that there are isomorphisms fµαβ Fµβ |Uαβ −−→ Fµα |Uαβ homqu commuting with the surjections. 3 these are unique and satisfy the cocycle condition fµαβ ◦ fµβγ = fµαγ on Uαβγ . Now the sheaves Fµα can be glued to global sheaves Fµ fµα −1 with isomorphisms Fµ |Uα −−→ Fµα such that fµαβ = fµα ◦ fµβ .

Em ⊂ V ⊗OX ′ and E1′ ⊂ . . ⊂ Em ⊂ V ⊗ OX isomorphic if there are isomorphisms Eµ ∼ = Eµ′ which are compatible with the inclusions and the identity on V ⊗ OX . Then we let F l(d, V )(X) be the set of isomorphism classes [E1 ֒→ E2 ֒→ . . ֒→ Em ֒→ V ⊗ OX ] of flags of subbundles of respective rank dµ . Using pullbacks we get a functor F l(d, V ) : (Var/k)op → (Sets). 1 says that this functor is representable and that the tautological flag is the universal object. 7. The full flag variety. Let F (V ) denote the flag variety F (1, .

Proposition: In the above notation G0 ×H0 ∼ = Hilb(P2 , 2m+2, 2m+1) ∼ = Hilb(P2 , 2m+2) and the projection onto H0 is the canonical projection onto Hilb(P2 , 2m + 1). Proof. Let Z ⊂ Z ⊂ S × P be any flat flag of subvarieties with Hilbert polynomial 2m + 1 and β 2m + 2. The family Z is induced by a unique morphism S − → H0 because H0 is the Hilbert scheme Hilb(P2 , 2m + 1) of conics in P2 . Then we have the diagram 62 0  0 C 0  / I / OS×P  / OZe / 0 0  / β ∗ OH0 (−1) ⊠ OP (−2) / OS×P  / OZ / 0   0 C  0 The ideal sheaf C is also flat over S because OZ and OZe are so.

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Algebraic Topology, Gottingen 1984 by L. Smith


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