BINARY QUARTICS AND THE EQUATIONS OF THE THIRD SECANT VARIETY OF THE GRASSMANNIAN Gr(4,8)

The third secant variety σ3(Gr(4,8)) ⊂ P(Λ^4 C^8) = P^69 is one of the few defective secant varieties of Grassmannians: it has dimension 49 instead of 50. Writing C^8 = C^2 ⊗ C^4, we show that its affine cone is the closure of the SL_8-orbit of the space S^4 C^2 of binary quartics, which is the fixed space of the generic stabilizer SL_4 and is spanned by a rational normal quartic curve of Slater determinants. We determine the ideal I of σ3(Gr(4,8)) in all degrees d ≤ 8 as a representation of SL_8:- I_d = 0 for d ≤ 4;- I_5 is the module dual to V(2ω2) plus its Hodge dual, of dimension 672;- I_6 = Δ*·I_5 (no new generators);- in degree 7 the modules dual to V(4ω1) and V(4ω7), of dimension 330 each, are new minimal generators;- there are no new generators in degree 8; in particular the invariant octic R_8 vanishing on the secant variety lies in the ideal generated by the quintics.The complete decomposition of I_d and the Hilbert function are given for d ≤ 8. Restricting covariants to the binary quartics gives a lower bound for the number of equations in each isotypic component in terms of covariants of the binary quartic. This bound accounts for all minimal generators of degree ≤ 8, and the invariants of the θ-representation Λ^4 C^8 of E_7 restrict to the classical invariants I and J^2. We give explicit formulas for the generators: the quintic covariant is the curvature-tensor part of 36 A_T^2 M_T − 5 (tr A_T) A_T M_T, where A_T is Katanova's quadratic operator on Λ^2 C^8, and the septic covariant is a single contraction of four copies of T with three copies of its Hodge dual. The proofs combine a stabilizer criterion, an extremal projector of Asherova–Smirnov–Tolstoy and rank certificates computed modulo a prime, used only in the direction in which they are valid over Q. In fermionic quantum information, the equations are necessary conditions for a state of four fermions in eight modes to have Slater border rank at most 3. The record includes the preprint and the supplementary material: all programs (Python/NumPy), their complete output, the exact integral bases of the quintic modules and the evaluation points used in the certificates.