The varieties of subspaces stable under a nilpotent transformation

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Let f : V → V be a nilpotent linear transformation of a vector space V of type V = λ, i.e. the size of Jordan blocks λ_1 ≥ λ_2 ≥ ･･･ ≥ λ_1. For an f-stable subspace W of V, i.e. f(W) ⊂ W, the types of W and V/W are those of the maps f|w : W → W and fv/w : V/W → V/W induced by f, respectively. For partitions νand μ we investigate the set S(λ, ν, μ) = {W ⊂ V; f(W) ⊂ W, type W = ν, type V/W = μ} and the singular locus of the Zariski closure X(λ, ν, μ) of S(λ, ν, μ) in the grassmaniann of subspaces of V of dimension |ν|. We show that S(λ, ν, μ) is nonsingular and its connected components are rational varieties (Th.A) ; generic vectors are introduced (Def.18), which define the generic points of the irreducible components of X(λ, ν, μ) whose Plücker coordinates are fairly simple to express their defining equations. We describe explicitly the coordinate ring of an affine openset of X(λ, ν, μ) with the singular locus of codimension two (Prop.C).

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  • Ryukyu mathematical journal

    Ryukyu mathematical journal 16 43-71, 2003-12-30

    Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus

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