Ternary Cubics, Four Qubits and a Local Obstruction to Normality of Secant Varieties in E8

We give two criteria that detect non-normality of a variety X in a θ-representation whose affine cone is the closure of the orbit of a subspace a of a Cartan subspace. Typical examples are the secant varieties of closed orbits.- Global criterion: the invariant ring of the cone is the image of the invariants of the little Weyl group W. Its integral closure is the invariant ring of the relative group W_a. If W_a has more basic invariants of some degree than W, then X is not projectively normal.- Local criterion: if the local group of P(a)/W_a at a point [q] has more basic invariants of some degree d than the parabolic subgroup Z_W(q) has invariants of degree d, then X is not normal at [q]. Applications to the two θ-representations of e8:- Gr(3,9) in P(Λ^3 C^9), little Weyl group G32. The cone over the fourth secant variety is the orbit closure of the ternary cubics, with relative group G5, and we determine its invariant ring in terms of the Aronhold invariants. σ3 is not projectively normal, and σ4 is not normal along σ3.- The spinor variety S16 in P^127, little Weyl group W(E8). The cone over σ4 is the orbit closure of the four-qubit space, with relative group W(F4). σ3 is not projectively normal, and σ4 is not normal along σ3.- Contrast: three qutrits P^2×P^2×P^2 have the same relative groups and the same local group as Gr(3,9), but neither criterion applies. We prove that Strassen's hypersurface σ4 is normal, so every secant variety of P^2×P^2×P^2 is normal and projectively normal. Survey: both criteria are evaluated on all flats of W(G2), W(F4), W(E6), W(E7), W(E8), G25 and G32, and on all non-flat eigenspaces of dimension at least 2 of these groups other than W(E7) and W(E8). This gives 118 subspaces and 1073 strata.- On flats the global test is sharp: the restriction of invariants is surjective exactly when W_a is a reflection group whose degrees, with multiplicity, are degrees of W. This agrees with the criterion of Amend, Berardinelli, Douglass and Röhrle in all 103 cases where it was checked.- The local test, with a degree d ≥ 2 that is not a sum of degrees of the parabolic subgroup, applies at 32 strata. In the Weyl groups d = 3 in all but one of them. Methods. Exact arithmetic (integers, rationals, cyclotomic fields) is used for every statement except two kinds ofcomputation:- Ranks modulo a prime, used only as lower bounds.- Floating point, used only to search for candidates, which are then verified exactly.For W(E7) and W(E8) the survey enumerates the relative groups as permutation groups, without enumerating W. It is cross-checked against the exact computation for W(E6).