Classification of products of Fano varieties with Picard number one

Abstract

Given a partition $(n_1,\ldots,n_r)$ of a positive integer $n$, one has the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. We show that distinct partitions of $n$ yield non-isomorphic multiprojective spaces, giving a new proof via the extremal contractions of their closed cone of curves. In contrast to the earlier approaches, the argument here is uniform across all partitions, and extends beyond multiprojective spaces. In fact, we further extend it to a more general setting, namely to products of Fano varieties of Picard number one: we prove that a fixed such factor in each dimension makes the products attached to distinct partitions pairwise non-isomorphic. As a consequence, a complete classification of products of smooth quadrics, of dimension $\geq 3$, has been obtained.