Higher dimensional polarized varieties with non-integral nefvalue

Let X be an n-dimensional normal projective variety with terminal, Gorenstein, Q-factorial singularities. Let L be an ample line bundle on X. Let tau be the nefvalue of (X; L). Then we classify (X; L), describing the structure of the nefvalue morphism of (X; L), when tau satisfies n - k < tau < n - k + 1 and n greater than or equal to 2k - 3, k greater than or equal to 4. In the smooth case, we discuss the case n = 2k - 4, k greater than or equal to 5, as well.