Projective threefolds of small class (Q1820837)

Let \(X\subset {\mathbb{P}}^ n\) be a nondegenerate complex projective threefold of degree \(d\) and let \(\mu\) be its class, i.e. the number of hyperplanes tangent to X contained in a general pencil. By combining the class formula, expressing \(\mu\) in terms of the Betti numbers of X and of its linear sections, with recent results on surfaces and threefolds with low sectional genus, the authors determine the X as above with \(\mu \leq 8\). Moreover, restricting to a particular class of threefolds, which are \({\mathbb{P}}^ 1\)-bundles over a smooth surface, the authors classify those X for which \(\mu-d\) is small.