Automorphism groups of spaces of minimal rational curves on Fano manifolds of Picard number 1

Abstract

Let X be a Fano manifold of Picard number 1 and M an irreducible component of the space of minimal rational curves on X. It is a natural problem to understand the extent to which the geometry of X is captured by the geometry of M. In this vein we raise the question as to whether the canonical map Aut o(X) → Auto (M) is an isomorphism. After providing a number of examples showing that this may fail in general, we show that the map is indeed an isomorphism under the additional assumption that the subvariety of M consisting of members passing through a general point x ∈ X is irreducible and of dimension ≥ 2.