Linear recurrence sequences and the duality defect conjecture

TL;DR

This paper explores the duality defect conjecture for smooth subvarieties in projective space, providing a combinatorial approach using recurrence sequences and proving the conjecture for certain cases.

Contribution

It generalizes an existing algorithm to higher codimensions and establishes the conjecture for codimension 3 when the ambient space dimension is odd.

Findings

Proves the duality defect conjecture for codimension 3 in odd-dimensional projective spaces.

Develops bounds on degrees of potential counterexamples using recurrence sequences.

Extends combinatorial algorithms to higher codimension cases.

Abstract

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space 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 proving the conjecture in the codimension $2$ case was developed by Holme, and following this approach Oaland devised an algorithm for proving the conjecture in the codimension $3$ case for particular $N$. This combinatorial approach gives a potential method of proving the duality defect conjecture in many of the 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 with this bounds the degrees of counterexamples would 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.