# A Galois theory example by Brian Osserman By Brian Osserman

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Extra resources for A Galois theory example

Example text

Summing up, we obtain the following result. 8. Theorem. If a variety X n ⊂ PN corresponds to the orbit of highest weight vector of an irreducible representation of a simple Lie group G and dim SX < N ≤ 2n + 1, then X is one of the seven varieties A1 )–A4 ), C), E), F). The varieties A1 ), A3 ), and E) are Severi varieties, and the varieties A0 ), A2 ), C), and F) are hyperplane sections of Severi varieties. 8 we obtain the following result. 9. Theorem. Over an algebraically closed field K of characteristic zero there exist exactly four Severi varieties corresponding to orbits of linear actions of algebraic groups, viz.

E. codimPN X n = N − n ≥ 2N ≥ 2s + 2 ≥ 2n + r − b + 2, 1 (r − b) + 1. 2 42 II. 2. Corollary. Let Y r ⊂ X n ⊂ PN , where X is nonsingular in a neighborhood of Y . Suppose that there is a point u ∈ PN \ X such that the projection π : X → PN −1 with center at u is a closed embedding in a neighborhood of Y . Then N ≥ n + 12 (r + 3). 3. Remark. 1 (resp. 2) is true if instead of assuming that π is a J-embedding with respect to Y (resp. an embedding in a neighborhood of Y ) we assume that π is J-unramified with respect to Y (resp.

Gv and 0, and therefore all Λi are collinear (cf. , ). On the other hand, since Gv is a cone, all Λi lie in an affine hyperplane (cf. ). Thus we may assume that r = 1 and Gv is the orbit of highest weight vector of an irreducible representation of a semisimple group G (varieties of such type were considered in  and were called HV -varieties). In particular, from this it follows that the variety X = Gv/K ∗ ⊂ PN is rational (cf. 3 below) and is defined in PN by quadratic equations (cf.