The critical space for orthogonally invariant varieties [article]

Giorgio Ottaviani
2021 arXiv   pre-print
Let q be a nondegenerate quadratic form on V. Let X⊂ V be invariant for the action of a Lie group G contained in SO(V,q). For any f∈ V consider the function d_f from X to C defined by d_f(x)=q(f-x). We show that the critical points of d_f lie in the subspace orthogonal to 𝔤· f, that we call critical space. In particular any closest point to f in X lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of
more » ... , Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety.
arXiv:2104.14998v1 fatcat:nihq5wn2srhq3lki7j5fmidqze