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By Garrett Birkhoff

This vintage, written through younger teachers who grew to become giants of their box, has formed the certainty of contemporary algebra for generations of mathematicians and is still a beneficial reference and textual content for self examine and school classes.

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The representation of HΛ in V obtained by restricting the representation of G is reducible: it is clear that HΛ · (KvΛ ), HΛ · (gvΛ ) = gvΛ , where gvΛ is the tangent space to GvΛ at the point vΛ . 1) where KvΛ ⊕ t V = gvΛ is the tangent and n V is the ‘normal’ subspace to GvΛ ⊂ V at the point vΛ ∈ GvΛ . It is clear that the subset GvΛ ∩ (t V ⊕ n V ) is stable with respect to the action of HΛ , and the stabilizer of an arbitrary point of GvΛ ∩ (t V ⊕ n ) contains a maximal unipotent subgroup of HΛ .

1) and a linear subspace L ⊂ PN , dim L = N − N − 1 such that X is nondegenerate and if π : PN PN is the projection with center L, then π(X ) = X. From our assumptions on the dimension and degree of X it follows that L ∩ X = ∅. We may assume that N = N + 1. In fact, if N > N + 1, then we pick a general linear subspace L ⊂ L, dim L = N − N − 2 and denote by π : PN PN +1 the projection with center L . Put X = π (X ), L = π (L), and let π : PN +1 PN be the projection with center L (here L is a point in N +1 P ).

2]). 19. Remark. A variety X n ⊂ PN is called projectively normal if all its Veronese embeddings vk (X), k ≥ 1 are linearly normal (or, in other words, if the restriction maps H 0 PN , OPN (k) → H 0 X, OX (k) are surjective for all k ≥ 1). Rao [69] constructed threefolds in P5 which are not projectively normal. We already observed that all such varieties are linearly normal. CHAPTER III VARIETIES OF SMALL CODIMENSION CORRESPONDING TO ORBITS OF ALGEBRAIC GROUPS Typeset by AMS-TEX 47 48 VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS 1.

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