Adjunction and singular loci of hyperplane sections. II (Q743950)

Let \((X,L)\) be a smooth complex polarized variety of dimension \(n\) such that there exists an effective irreducible divisor \(A\in |L|\), and let \(\Sigma:=\text{Sing}(A)\) be the singular locus of \(A\) with \(k=\dim \Sigma\) and \(c=\text{codim}_{X}(\Sigma)\). In [\textit{M. C. Beltrametti} et al., J. Math. Soc. Japan 67, No. 2, 861--875 (2015; Zbl 1319.14007)], the authors investigated positivity conditions for adjoint bundles \(K_{X}+tL\) with \(t\geq n-3\) under the assumption that (1) \(\Sigma\) consists of non-degenerate quadric singularities and (2) \(\Sigma\) is smooth. More precisely, the authors studied the following: (a) The existence of the first reduction of \((X,L)\) under the assumption that \(k\geq 2\) and \(c\geq 3\). (b) The existence of the second reduction of \((X,L)\) and the nefness and bigness of the third adjoint bundle under the assumption that \(k\geq 3\) and odd codimension \(c\geq 3\). In this paper, the authors point out that the same analysis also works without the above assumption (1) by using a result of [\textit{P. Aluffi}, Duke Math. J. 80, 325--351 (1995; Zbl 0876.14028)]. In any case, however, the assumption that \(K_{\Sigma}+(k-2)L_{\Sigma}\) is nef is crucial when we study (b) above. So in this paper, the authors study the positivity of adjoint bundles on \(\Sigma\) with odd codimension \(c\), and they describe all \((\Sigma, L_{\Sigma})\) which satisfy one of the following types: (i) \(k\geq 2\) and \(K_{\Sigma}+(k-1)L_{\Sigma}\) is not ample. (ii) \(k\geq 3\), \(K_{\Sigma}+(k-1)L_{\Sigma}\) is ample but \(K_{\Sigma}+(k-2)L_{\Sigma}\) is not nef. (iii) \(k\geq 3\), \(K_{\Sigma}+(k-2)L_{\Sigma}\) is nef but not big. Moreover the authors give several explicit examples, and these show that the lists are effective.