On an Unusual Conjecture of Kontsevich and Variants of Castelnuovo’s Lemma

Let A = (a j ) be an orthogonal matrix with no entries zero. Let B = (b j ) be the matrix defined by b j = 1 ai 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 in algebraic geometry. §1. Definitions and Statements Definition 1.1. Given a k × l matrix A = (aα), 1 ≤ i ≤ k, 1 ≤ α ≤ l, with no entries zero, define the Hadamard inverse of A, B = (bα), by b i α = 1 aα . (The name is in analogy with the Hadamard product.) Maxim Kontsevich conjectured the following: Conjecture 1.2. (Kontsevich (1988)) Let A be an orthogonal matrix (either over R or C), with no entries zero. Let B be the Hadamard inverse of A. Then the rank of B is never equal to three. At first glance, (1.2) may not appear all that striking because based on a naive count, one would not expect any Hadamard inverses of low rank to exist. However Kontsevich asserted and we show the following: Theorem 1.3. The space of k × k orthogonal matrices with rank two Hadamard inverses is (2k − 3)-dimensional. We will rephrase (1.2), (1.3) in geometric language and prove them. First we will need some definitions: 1991 Mathematics Subject Classification. 114E07, 14M210, 115A99.