Ternary Cubics, Four Qubits and a Local Obstruction to Normality of Secant Varieties in E8
Year: 2026
Type: preprint
Abstract: 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, and its integral closure is the invariant ring of the relative group W_a. If W_a needs more generators in some degree than W, or has a larger Hilbert function, then X is not projectively normal.- Local criterion: if the local group H_q 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]. Structure and classification:- At a stratum of a flat, H_q contains the relative group of a flat inside the parabolic subgroup Z_W(q) as a normal subgroup with cyclic quotient. The quotient is trivial when -1 is in W. In this sense the local obstruction is the global one, one level down.- For Weyl groups containing -1, and for W(E6), this gives a complete classification of where the local criterion applies with a degree d >= 2 that is not a sum of degrees of the parabolic subgroup: - it never applies in types B_n, C_n, F4 and G2; - in types D_2k, E6, E7 and E8 it applies exactly through components of the parabolic of type E6 (then d = 3) or of type D_n with n odd (then d is given by an explicit block rule); - the degree d can be arbitrarily large (d = 2k - 3 in type D_2k). 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. In the classification, this is the pair (E6, D4).- 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. It agrees with the criterion of Amend, Berardinelli, Douglass and Röhrle in all 103 cases where it was checked.- The classification predicts the local obstruction correctly at all 847 strata of W(E7) and W(E8). Methods. Exact arithmetic (integers, rationals, cyclotomic fields) is used for every statement except two kinds of computation:- Ranks modulo a prime, used only as lower bounds.- Floating point, used only to search for candidates, which are then verified exactly.The single exception is the part of the survey concerning W(E7) and W(E8). There integral characteristic polynomials are obtained by rounding, and the results are cross-checked against the exact computation for W(E6). That part is not used in the proofs of the main theorems.
Author: TORREGROSA JIMÉNEZ, JOSÉ
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225450840