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

Abstract

Let $A=(a_j^i)$ be an orthogonal matrix with no entries zero. Let $B=(b_j^i)$ be the matrix defined by $b_j^i=\frac{1}{a_j^i}$. 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.

arXiv commentamstex

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