On the Fourth Adjoint Contractions of Divisorial and Fiber Types

In this paper, we will list up all the cases for the ray contractions of divisorial and fiber types for smooth projective varieties of dimension five. These are obtained as a corollary from the lists of n-dimensional k-th adjoint contractions f : X → Y of the same types for k = 1, 2, 3 and 4 (n ≥ 5). The lists for k = 1, 2 and 3 have previously been obtained in [Na, Proposition 1.2 and Theorem 1.3]. The main task will be to have such a list for k = 4, where one case in the list fails to show that a positive-dimensional general fiber F of f is irreducible when n > 5. This assertion will, however, be proven when n = 5 with an essential aid of 3-dimensional Minimal Model Program in [Mo2]. (We do not show the existence of cases.)