Duale Varietäten von Fahnenvarietäten
(joint work with Gisela Menzel)
Let V be an irreducible representation of a semisimple
algebraic group G and X the orbit of an highest weight
vector in the projective spave P(V). Then X is a
generalized flag variety. The dual variety of X is the set of
hyperplanes in P(V) which are tangent to X.
The purpose of the paper is to determine all cases in which the dual
variety is not of codimension one. When G is simple there are
the following cases (upto isomorphims of G and V):
G=SL(n,k), V=kn;
G=Spn, V=kn
(n even); G=SL(n,k),
V=\wedge2kn (n odd);
G=Spin(9) or Spin(10), V=k16.
The
classification for semisimple groups is easily obtained from that.
Appeared in: Commentarii Mathematici Helvetici
62 (1987) 38-61.
Available files:
Remarks:
- Meanwhile, this classification has been
simplified by Dennis Snow, The nef value and defect of homogeneous line
bundles. Transact. AMS, 340 (1993) 227-241.
- A careful analysis of the
hyperplane sections of X where V is the adjoint representation, was the
subject of another paper.
Back to index
Last updated: December 19, 2003