On the classification of the third reduction with a spectral value condition (Q1127006)

Let \(X_0\) be an \(n\)-dimensional smooth projective variety (\(n\geq 9\)) and let \(L_0\) be a very ample line bundle on \(X_0\) whose spectral value is \(>2[{n-1 \over 3}]\). It is known that if \(K_{X_0}+(n-1)L_0\) is nef and big there is a first birational contraction \(\Phi_1 :X_0 \to X_1\) onto a smooth projective variety \(X_1\) and \(L_{1} =:(\Phi_{1*}L_{0})^**\) is ample. If also \(K_{X_1}+(n-2)L_1\) is nef and big there is a second contraction \(\Phi_2 :X_1 \to X_2\) onto a normal projective variety \(X_2\); moreover if \(2[K_{X_2}+(n-3)L_2]\) is nef and big, where \(L_{2} =:(\Phi_{2*}L_{1})^**\), there is a third birational contraction \(\Phi_3 :X_2 \to X_3\) onto a normal projective variety \(X_3\). \(\Phi_1\) and \(\Phi_2\) have been studied by Beltrametti, Sommese and Fujita. In this note the author studies the singularities of \(X_3\) and classifies the exceptional locus of \(\Phi_3\).