Full Hodge-theoretic determination of the geometric Picard group

Determine whether the geometric Picard rank of the minimal resolution of the surface V parameterizing 3 × 3 magic squares of squares is maximal, namely whether it equals 544, and construct the remaining divisor classes if so.

Background

The paper constructs 1204 divisors on the resolution of V and proves that they generate a sublattice of rank 518. Hodge-theoretic considerations provide the upper bound 544 for the geometric Picard rank.

The authors state that they expect the rank to be maximal but explain that their explicit divisor constructions appear to be saturated. They therefore indicate that the remaining classes, if they exist, will likely require Hodge-theoretic methods. Because the passage uses “expect” rather than one of the requested uncertainty markers, it is not included under the strict quotation criterion unless the accompanying unresolved content is stated explicitly.