Sharp FRS and singularity thresholds for off-diagonal Gram maps

Abstract

The off-diagonal Gram map for h vectors in an M-dimensional quadratic space is flat with reduced fibers of rational singularities (FRS) in characteristic zero if and only if M > max_{2 <= r <= h} (2h - r + 1 - 2h/r). Equivalently, this gives the exact FRS convolution threshold of x |-> (x_i x_j){i < j}. For its zero-fiber ideal I{M,h}, we also prove lct(I_{M,h}) = min{ binom(h,2), min_{0 <= k <= h-2} ( binom(k+1,2) + M(h-k)/2 ) }. In the strict range, all truncated coefficient maps have geometrically integral complete-intersection fibers and are faithfully flat over Z[1/2], uniformly over graphs on h vertices. The proof uses a sharp weighted estimate for hollow symmetric matrices over finite local principal ideal rings, paired with isotropic lower bounds. Within the established point-counting approach to rational singularities, this estimate also gives an explicit modulus of continuity for Gram-map densities over every local field of odd residue characteristic.