Linear recurrence sequences and the duality defect conjecture

It is conjectured that the dual variety of a smooth nonlinear varietyX subset of PNof dimensiondim(X)>2N3is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of Hartshorne's complete intersection conjecture but nevertheless remains open for the case of subvarieties of codimension>2. A combinatorial approach to prove the conjecture in the codimension 2 case was developed by Holme, and following this approach Oaland devised an algorithm to prove the conjecture in the codimension 3 case for particularN. This combinatorial approach gives a potential method to prove the duality defect conjecture in many cases by studying the positivity of certain homogeneous integer linear recurrence sequences. We give a generalization of the algorithm of Oaland to the higher codimension cases, obtaining bounds that the degrees of counterexamples to the duality defect conjecture have to satisfy, and using the relationship with recurrence sequences we prove that the conjecture holds in the codimension 3 case whenNis odd.