Let $X$ be an $n$-dimensional (smooth) intersection of two quadrics, and let ${T^{\rm{*}}}X$ be its cotangent bundle. We show that the algebra of symmetric tensors on $X$ is a polynomial algebra in $n$ variables. The corresponding map ${\rm{\Phi }}:{T^{\rm{*}}}X \to {\mathbb{C}^n}$ is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, which is a quotient of a hyperelliptic Jacobian by a $2$-torsion subgroup. In dimension $3$, ${\rm{\Phi }}$ is the Hitchin fibration of the moduli space of rank $2$ bundles with fixed determinant on a curve of genus $2$.

In the Mathematical Review of ‘On a conjecture of Kontsevich and variants of Castelnuovo's lemma’ [Compositio Mathematica 115 (1999), 205–230], Gizatullin pointed out that the proof as written is incomplete for the complex case because the possibility of a certain equation vanishing identically was ignored. This corrigendum addresses the missing case. Note that it is simply another iteration of the argument already present in the article.

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/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 statement can be understood as variants of the Castelnuovo lemma and Brianchon‘s theorem.