On the class of some projective varieties
Recall that the class of an integral variety \(Y\subset \mathbb{P}^N\) is the degree of the dual variety \(Y^* \subset \mathbb{P}^{N^*}\) if \(Y^*\) is a hypersurface, 0 otherwise. Here we prove some inequalities between the class and the degree of a smooth complex projective manifold and apply them to particular classes of varieties (e.g. scrolls and varieties with low sectional genus). For instance we prove that for a smooth linearly normal complex variety \(X \subset \mathbb{P}^N\) with degree \(d\), class \(\mu\) and dimension \(r\) (hence codimension \(c:= N-r)\) we have \(\mu\leq d(2d+ c-2 )^{2r}/ (4cd)^r \).
scientific article; zbMATH DE number 937446