Minimal Nodal Complexity for Detecting the Primitive Middle Hodge Class on a Smooth Quadric Fourfold

Abstract

This preprint studies a concrete generalized-Thomas-section benchmark on the smooth quadric fourfold Q^4. For the primitive middle Hodge class ζ=2[Π]−h^2 associated with a maximal plane Π, it proves—subject to independent verification and without a claim of novelty—that the minimum number of ordinary double points on a reduced divisor D∈|O_Q(2)| with ζ|_D≠0 is three. The lower bound combines Kloosterman's defect analysis for nodal (2,2) complete intersections with Totaro's identification of the divisor class group of a klt Fano variety with ordinary homology; the upper bound is given by an explicit three-node quadric section containing Π. The result is a local benchmark in Hodge-theoretic detection geometry and is not claimed as progress on the Hodge Conjecture itself.