6.1. The direction complex and its degeneracy loci
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Fix a root o \in V, set n_0 = |V| - 1 and m = |E_F|, and let
X^\circ_{V,o} be the space of configurations with a_o = 0 and pairwise
distinct points, a smooth irreducible open variety of dimension 3n_0. The
direction complex C^\bullet_F is the two-term complex of trivial bundles on
X^\circ_{V,o} given by \partial_F(a) = D_{F,o}(a)^T, the transpose of
the rigidity matrix with the root's three columns deleted; for
1 \le s \le m the sth stress-degeneracy subscheme \Sigma_s(F) is the
vanishing of the determinantal ideal sheaf of minors of order m - s + 1 of
\partial_F, so its points are the configurations whose self-stress space
has dimension at least s.
(Zheng, 2026, Equation 4.2–4.3)
The determinantal construction is the standard one for a morphism of vector
bundles ((Bruns and Vetter, 1988);
(Fulton, 1998)). The section illustrates it on the triangle:
by Example 4.1 of Zheng (2026), three pairwise distinct
collinear points give K_3 a one-dimensional self-stress space, and the
locus of such configurations has codimension two in X^\circ_{V,o}, so
\codim \Sigma_1(K_3) = 2 \ge 1.