On the adjunction theoretic classification of projective varieties
Let \(X^{\wedge}\) be a connected projective submanifold of \({\mathbb{P}}_{{\mathbb{C}}}\) of dimension \(n\geq 3\), and let \(L^{\wedge}={\mathcal O}_{{\mathbb{P}}_{{\mathbb{C}}}}(1)_{X^{\wedge}}\). In this article we give a precise biregular structure theory for those pairs \((X^{\wedge},L^{\wedge})\) with \(n\geq 6\), such that \(| m(K_{X^{\wedge}}+(n-3)L^{\wedge})|\) does not give a birational map for any positive \(m.\) For \(n=3, 4, 5\), we give partial results, i.e. we give a precise biregular structure theory for those pairs \((X^{\wedge},L^{\wedge})\) such that \(| m(K_{X^{\wedge}}+tL^{\wedge})|\) does not give a birational map for any positive m with mt integral and \(t=2/3, 3/2\), and 9/4 for \(n=3, 4, 5\) respectively. In the case of \(n=4\) these results were obtained by \textit{M. L. Fania} and the reviewer [Ark. Mat. 27, No.2, 245-256 (1989; Zbl 0703.14005)]. Some simple applications of these results are given. Let \((X^{\wedge},L^{\wedge})\) be such that \(n\geq 6\), and assume that the degree of \(X^{\wedge}\) embedded by \(| L^{\wedge}|\) is \(>g(L^{\wedge})-1,\) where \(g(L^{\wedge})\) is the genus of a curve section of \(X^{\wedge}\). Then \(| m(K_{X^{\wedge}}+(n- 3)L^{\wedge})|\) does not give a birational map for any positive \(m,\) and we can apply our result to get precise information on the biregular structure for these pairs. Some applications are also made to pairs \((X^{\wedge},L^{\wedge})\) with the discriminant locus, \({\mathcal D}\), not a divisor on \(| L^{\wedge}|\).
scientific article; zbMATH DE number 4171104
scientific article