By Ciro Ciliberto, Vincenzo Di Gennaro (auth.), Vladimir L. Popov (eds.)
"... This publication provides an outstanding flavour of a few present study in algebraic transofrmation teams and their purposes. ..."
B.Martin, publication of the hot Zealand Mathematical Society, No. ninety three, April 2005
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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
HM1], Proposition 10) For any plane Λ ⊂ Wx corresponding to a tangent line to the smooth locus of Cx , there exists a local analytic integral surface of the distribution W which contains x as a smooth point and has Λ as its tangent space at x. In particular, the Frobenius bracket tensor [ , ]x : ∧2 Wx → Tx (X)/Wx at a generic point x ∈ X annihilates any element corresponding to tangent lines to the smooth locus of Cx . 2. ([HM4], Proposition 4) At a generic point x ∈ X, the symbol algebra of W has dimension n = dim(X).
Remark. Here and in what follows, whenever we identify an irreducible SL(V )-module or an irreducible SL(W )-module with another, the identiﬁcation is unique up to a choice of multiplicative constant by Schur lemma. We will implicitly make a ﬁxed choice of the identiﬁcation. The projective geometry of Z ⊂ PU we are interested in is about the linear 2 span of the variety of tangent lines of Z in U. 2. The tangent lines to S, regarded as points of P( 2 U1 . Pl¨ ucker map, span Proof. The irreducible K-module decomposition of 2 2 2 (V ⊗ W ) = ( U1 is 2 V ⊗ S W ) ⊕ (S V ⊗ 2 2 Consider a plane in S of the form C(v1 ⊗ w) + C(v2 ⊗ w) W ).
1 − a1 1 − a1 Now notice that there is a positive integer N such that am a2 w2 λ + · · · + wm λ 1 − a1 1 − a1 am a2 w2 N λ + · · · + wm N λ ∈ P, = 1 − a1 1 − a1 μ ˜=N is a convex combination of m − 1 W -translates of N λ. Thus also every W translate of μ ˜ has the same properties and, by the inductive hypothesis, if we ˜ which lies in P+ , we obtain that denote by μ the unique W -translate of μ there exists a M such that VMμ is an irreducible component of VN⊗M λ . But 24 Corrado De Concini ⊗MN we know that VN⊗M , so that VMμ is an irreducible λ is a summand of Vλ ⊗MN .
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.)