A bound on the plurigenera of projective varieties (Q1416705)

Let \(X\) be a smooth projective variety over the complex field, embedded in the projective space \(\mathbb P^r\) and non-degenerate. J. Harris showed that the geometric genus of \(X\) is bounded by a constant which depends on \(r\), \(n=\dim(X)\) and \(d=\deg(X)\), and varieties attaining the maximal value are classified. In the paper under review, the author finds a similar bound for the plurigenera of \(X\), i.e. for the invariants \(p_i(X)=h^0(i\omega_X)\), where \(\omega_X\) is the canonical class. The author restricts himself to varieties of dimension \(n<r/2\) and degree \(d\) big with respect to \(r\). Also he works only for \(i\)-th plurigenera with \(i\gg r\). These restrictions are needed in the proof, for they allow a reduction to subvarieties of some variety of minimal degree. In the previous range, the bounds are sharp and varieties for which \(p_i(X)\) is maximal are classified. It turns out that also some varieties with non-minimal geometric genus have minimal plurigenus \(p_i(X)\) for some \(i>1\).