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

"... This ebook provides a very good flavour of a few present study in algebraic transofrmation teams and their purposes. ..."

*B.Martin, e-newsletter of the recent Zealand Mathematical Society, No. ninety three, April 2005*

**Read 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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**Extra info 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**

**Example text**

Ih |j1 , . . , jh ] the determinant of the minor of Y formed by the rows i1 , . . , ih and the columns j1 , . . , jh . We can then identify H 0 (X, Lω2 ) with the subspace of F[zi,j ] spanned by 2 × 2 minors of Y and H 0 (X, L2ω2 ) with the span of the polynomials [s, t|s , t ][u, v|u , v ] and [s, t, u|s , t , u ][v|v ] with 1 ≤ s, t, u, v, s , t , u , v ≤ 4. The image of H 0 (X, Lω2 ) ⊗ H 0 (X, Lω2 ) in H 0 (X, L2ω2 ) is the the span of the polynomials [s, t|s , t ][u, v|u v ]. A direct computation (see [Br] section 4 and 5), shows that the polynomial [2, 3, 4|2, 3, 4][1|1] does not belong to this image.

N, xj is the image of the tj in R1 /R2 . (3) Let λ ∈ Pic(Y) be a dominant weight. Then the ring Rλ = H 0 (Y, Lnλ ) n≥0 is normal with rational singularities. Proof. The proof of (1) is identical to that given in [DCP] for the case of X. 3. 2, the restriction of the map h∗ : M = H 0 (G/B, Lλ ) → H 0 (G/B × G/B, Lλ|G/B×G/B ) to M U is an isomorphism for all dominant λ ∈ P˜+ . 6. Let λ, μ ∈ Pic (Y) ∩ P˜+ , then the multiplication map m : H 0 (Y, Lλ ) ⊗ H 0 (Y, Lμ ) → H 0 (Y, Lλ+μ ) is surjective.

S ) as the closure of GS ⊂ M . 12. 1) The variety Y (v1 , . . , vs ; Ω1 , . . , Ωs ) is normal with rational singularities if and only if Ωi = Ω(λi ) for each i = 1, . . , s. 2) For a general sequence Ω1 , . . , Ωs , with Ωi ⊂ Ω(λi ), the normalization of Y (v1 , . . , vs ; Ω1 , . . , Ωs ) is given by Y (v1 , . . , vs , Ω1 (λ1 ), . . , Ωs (λs )). In particular, Y (v1 , . . , vs ; {λ1 }, . . , {λs }) is normal if and only if λi is minuscule for each i = 1, . . , s. Proof. Let Γ ⊂ T be the intersection of the kernels of the characters λi .