2018/01/31 by Grayson Jorgenson
Computer Science · Mathematics · #Codimension #Commutative Algebra and Its Applications #Conjecture #Counterexample #Dimension (graph theory) #Duality (order theory) #Generalization #Polynomial and algebraic computation #Tensor decomposition and applications #Variety (cybernetics) #math.AG
paper · pdf · doi:10.1112/blms.12379
Minor improvements to exposition. Submitted
openalex created_date 2018/01/26 · arxiv created 2018/02/19 · openalex publication_date 2020/06/23 · arxiv updated 2020/07/01 · openalex updated_date 2026/08/05
It is conjectured that the dual variety of a smooth nonlinear variety X ⊆ P N of dimension dim ( X ) > 2 N 3 is 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 particular N. 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 when N is odd.