Rationality of instanton moduli

We consider the moduli space $I_n$ of rank-2 mathematical instanton vector bundles with second Chern class $n\ge1$ on the projective space $P^3$. The irreducibility, respectively, the rationality of $I_n$ was known for $n\le5$, respectively, for $n=1,2,3$ and 5. It was recently proved by A.Tikhomirov that $I_n$ is irreducible for all odd values of $n$. Now the question of rationality of $I_n$ for arbitrary $n$ is in order. In this talk we discuss the recent result of D.Markushevich and A.Tikhomirov answering the question of irreducibility of $I_n$. We give the proof of the following theorem. THEOREM. Whenever $I_n$ is irreducible, it is rational. In particular, $I_n$ is rational for all odd $n\ge1$ and for $n=2,4$.