On an unusual conjecture of Kontsevich and variants of Castelnuovo's lemma

TL;DR

This paper proves Kontsevich's conjecture that an orthogonal matrix with no zero entries cannot produce a matrix of rank three when taking reciprocals, generalizing classical geometric lemmas like Castelnuovo's.

Contribution

It provides a geometric interpretation and proof of Kontsevich's conjecture, extending classical lemmas to a new algebraic setting.

Findings

Confirmed that the rank of the reciprocal matrix is never three for such orthogonal matrices

Generalized Castelnuovo's lemma and Brianchon's theorem in a new algebraic context

Established a geometric interpretation of Kontsevich's conjecture

Abstract

Let $A=(a^i_j)$ be an orthogonal matrix with no entries zero. Let $B=(b^i_j)$ be the matrix defined by $b^i_j=\frac 1{a^i_j}$. M. Kontsevich conjectured that the rank of $B$ is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statment can be understood as a generalization of the Castelnouvo lemma and Brianchon's theorem.