On a conjecture of Kontsevich and variants of Castelnuovo's lemma
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Let A = (aij) be an orthogonal matrix (over ℝ or ℂ) with no entries zero. Let B = (bij) be the matrix defined by bij = 1/aij. M. Kontsevich conjectured that the rank of B is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statement can be understood as variants of the Castelnuovo lemma and Brianchon's theorem.
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