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Download e-book for iPad: Algebraic Transformation Groups and Algebraic Varieties: by Ciro Ciliberto, Vincenzo Di Gennaro (auth.), Vladimir L.

By Ciro Ciliberto, Vincenzo Di Gennaro (auth.), Vladimir L. Popov (eds.)

ISBN-10: 3642058752

ISBN-13: 9783642058752

ISBN-10: 3662056526

ISBN-13: 9783662056523

"... This e-book provides an excellent flavour of a few present learn in algebraic transofrmation teams and their functions. ..."

B.Martin, publication of the recent Zealand Mathematical Society, No. ninety three, April 2005

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Read Online or Download Algebraic Transformation Groups and Algebraic Varieties: Proceedings of the conference Interesting Algebraic Varieties Arising in Algebraic Transformation Group Theory held at the Erwin Schrödinger Institute, Vienna, October 22–26, 2001 PDF

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Additional resources for Algebraic Transformation Groups and Algebraic Varieties: Proceedings of the conference Interesting Algebraic Varieties Arising in Algebraic Transformation Group Theory held at the Erwin Schrödinger Institute, Vienna, October 22–26, 2001

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Let σ be a nonvanishing section of the invertible sheaf pr∗ F and let σ t be the corresponding section of F t on P1 . For t = 0, the section σ t of F t is nowhere vanishing on P1 . If σ 0 has a zero, we can easily get a contradiction to the assumption (a). Thus σ 0 is nowhere vanishing and σ defines a trivial line subbundle O of F with locally free quotient G = F /O. Let G t be the restriction of G on P1 × {t}. Then G t ∼ = O(−1)3 for all t = 0. Applying the same argument using the assumption (a) to pr∗ (G ⊗ O(1)), we see that G 0 ∼ = O(−1)3 .

1 ), in which case Zλ is the cone over the Segre embedding of (P3 )n . 4 A Generalization As in the previous section, G will denote a semisimple simply connected algebraic group over k, T ⊂ G a maximal torus and B ⊂ G a Borel subgroup containing T . We shall continue to denote by G the adjoint quotient of G. We now take another semisimple simply connected group G, a maximal torus T in G and a Borel subgroup B ⊃ T in G. We shall set P˜ = X(T ), the ˜ ⊂ P˜ denote the set of roots, and let Δ˜+ ⊂ Δ be character group of T .

Let us identify our base curve P1 with PW and the hyperplane line bundle O(1) with the dual tautological line bundle h on PW such that H 0 (PW, h) = W ∗ . Also make an identification of S with PV ∗ where V = V ∗ ⊗ h, so that the subspace of H 0 (Z, ξ)∗ spanned by S ⊂ PH 0 (Z, ξ)∗ is canonically identified with V ⊗ W . Under these identifications, H 0 (S, ξ) = H 0 (PV ⊗ h∗ , ξ) = H 0 (PW, V ∗ ⊗ h) = V ∗ ⊗ W ∗ in a canonical manner. The kernel of the restriction map H 0 (Z, ξ) → H 0 (S, ξ) can be naturally identified with the subspace of 2 E) H 0 (Z, ξ) = H 0 (PW, consisting of sections of the subbundle 2 H 0 (PW, 2 V⊂ 2 2 E on PW .

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Algebraic Transformation Groups and Algebraic Varieties: Proceedings of the conference Interesting Algebraic Varieties Arising in Algebraic Transformation Group Theory held at the Erwin Schrödinger Institute, Vienna, October 22–26, 2001 by Ciro Ciliberto, Vincenzo Di Gennaro (auth.), Vladimir L. Popov (eds.)


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