Nonexistence of canonical small resolutions for symmetric rank loci

The symmetric rank loci are varieties defined by rank conditions on symmetric matrices. A canonical small resolution means a small resolution defined canonically for such a locus.

Nonexistence conjecture. There do not exist canonical small resolutions for symmetric rank loci.

The conjecture is motivated by the expected jump in fiber Euler characteristics implied by the intersection-complex stalk Euler characteristics, in contrast with the behavior of ordinary matrix rank loci. Its status is not established in the supplied source.