The Critical Space for Orthogonally Invariant Varieties

Abstract

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 df from X to $\mathbb C$ ℂ defined by df(x) = q(f − x). We show that the critical points of df lie in the subspace orthogonal to ${\mathfrak g}\cdot f$ 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 Draisma, Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety.

The span of singular tuples of a tensor beyond the boundary format

“…In Definition 4.2 we call it critical space as in [DOT18,Definition 2.8]. Due to its relevance in Euclidean distance optimization, more recently Ottaviani [Ott22] defined critical spaces of algebraic varieties invariant for the action of a Lie subgroup of the orthogonal group. In few words, the equations of H T can be obtained from the equations defining singular k-tuples not restricted to rank-one solutions.…”