Birationally rigid Fano-Mori fibre spaces (Q6633889)

Let \(M \in \mathbb Z\), and let \(\mathcal P\) be the space of all tuples \(f = (f_1, \dots, f_k)\) of homogeneous polynomials of degrees \(d_1, \dots, d_k\) on \(\mathbb P^{M+k}\).\N\NTheorem 0.1: There is a Zariski-open subset \(\mathcal F \subset \mathcal P\) such that for \(f \in \mathcal F\), the zero scheme \(\{ f = 0 \}\) is an irreducible, reduced, factorial Fano variety with terminal (and ``divisorially canonical'') singularities. The codimension of the complement of \(\mathcal F\) grows at least quadratically in \(M\) when \(k\) is fixed.\N\NA Fano-Mori fibre space \(\pi \colon V \to S\) is a Mori fibre space with some additional properties, such as \(S\) being smooth and rationally connected, the fibre dimension being at least three and the Picard groups of \(V\) and of the fibres being as simple as possible.\N\NTheorem 0.2 gives a criterion for a Fano-Mori fibre space to be birationally rigid in terms of certain technical conditions.\N\NTheorem 0.3 provides a construction of Fano-Mori fibre spaces that actually satisfy those conditions.